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Geometry and Measurement
Plane Geometry - Properties of Basic Shapes (Points, Lines, Angles, Triangles, Quadrilaterals, Circles)
Q.01
'Find the value of k representing a circle passing through the origin (0,0).'
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'Understand the explanation of distance between 1 points. Find the formulas for the distance between the origin O and point P(a), and between points A(a) and B(b).'
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'(1) What kind of shape does the equation represent?\n(2) In order for the equation to represent a circle, determine the range of values for the constant .'
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'(1) Find the coordinates of the midpoint of the chord formed by the intersection of the line x+y=1 and the circle x^{2}+y^{2}=4, and determine the length of the chord.'
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'(1) Passing through both the x-axis and y-axis, and point A(-4,2). (2) Passing through point (3,4), touching the x-axis, and having its center on the line y=x-1.'
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'Plot the region that satisfies the inequalities and , and explain their relationship.'
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'Example 29 | Shape of a triangle For 4 points A(4,0), B(0,2), C(3,3), D, answer the following questions.'
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'Find the coordinates of the internal division points, external division points, and centroid in Example 30'
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'Practice (63=>This Book p.137) (2) Let the coordinates of point P be (x, y), then from AP^2+BP^2=18 we get {(x-1)^2+(y-4)^2}+{(x+1)^2+y^2}=18, simplifying gives x^2+y^2-4y=0, which implies x^2+(y-2)^2=2^2. Therefore, the points that satisfy the condition lie on the circle (1). Conversely, any point on the circle (1) satisfies the condition. Hence, the desired locus is a circle with center at (0,2) and radius 2.'
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'For the line segment connecting A (-3) and B (6), find the coordinates of the following points: (1) Point dividing internally in the ratio 2:1 (2) Point dividing externally in the ratio 2:1 (3) Point dividing externally in the ratio 1:2 (4) Midpoint'
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'Taking line BC as the x-axis and point P as the origin, the coordinates of the vertices of triangle ABC can be expressed as: \nA(a, b), B(-c, 0), C(2c, 0)\nwhere b ≠ 0, c > 0. Verify the equation 2AB² + AC² = 3(AP² + 2BP²).'
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'Find the equation of a circle that touches both the x-axis and the y-axis.'
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'Find the trajectories of the following points Q and R with respect to the point P moving on the parabola y=x^2 and the two points A(3,-1), B(0,2).'
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'Since the point (3,4) lies on the line 3x-2y-1=0, the desired line passes through the points (-7,-11) and (-1,6).'
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'Practice (2) Parabola y=x^2 and line y=m(x+2) intersect at different points A and B. Find the locus of the midpoint of segment AB as the value of m varies.'
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'The equation of the line passing through the points (-1,1) and (3,-1) is , which simplifies to .'
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'When the chapter 3 (28 t) takes all real values, for the three points A(t, t^{2}), B(t, t-2), C(t+√3, t^{2}-t-1), answer the following questions:\n(1) Prove that for each real number t, A and B are distinct points.\n(2) Find all t values that make triangle ABC a right triangle.\n(3) Determine the range of t values that make triangle ABC an acute triangle.'
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'The equation of the perpendicular bisector of line segment BC is y-0=-2(x-5), which simplifies to y=-2x+10. By solving equations (4) and (5) simultaneously, we get x=4, y=2. Therefore, the center of the circumscribed circle is at point (4,2) and the radius is sqrt{(8-4)^{2}+(5-2)^{2}}=5. Thus, the equation we are looking for is (x-4)^{2}+(y-2)^{2}=25.'
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'When the intersection points of 22 circles, a circle passing through the intersection points of a circle and a line, and the equations of lines in terms of x, y are written as f(x, y), the curve represented by the equation f(x, y) = 0 (including cases where it represents a line) is called the curve f(x, y) = 0 and the equation is called the equation of the curve.'
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'Find the equation of the tangent line at point P(4,6) on the circle.'
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'Find the values of the constant k for which the lines do not form a triangle.'
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'The results from (1) to (3) indicate that the coordinates of the centroid G of △PQR change from to .'
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'(2) Find the locus of the point P for which the sum of the squares of the distances from the points A(1,4) and B(-1,0), i.e., AP^2 + BP^2 is 18. [(2) Hokkai Gakuen University]'
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'Given that the length of the perpendicular dropped from point (2,1) to the line kx + y + 1 = 0 is √3, find the value of the constant k.'
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'Find the equations of the lines parallel and perpendicular to the line 4x+3y-6=0 passing through the intersection point of the lines 2x-y-1=0 and x+5y-17=0.'
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'Find the distance between the following two points.'
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'(2) (1) From (1), in triangle △AOB where ∠AOB = 90°, the circle passing through points A, B, O has AB as its diameter.'
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'Find the arc length and area of a sector with a radius of 4 and a central angle of 150°.'
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'Problem (1) In the coordinate plane, when the points A(a, 2), B(5, 1), C(-4, 2a) are collinear, find the value of the constant a.'
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'(1) It touches both the x-axis and the y-axis, passing through point A(-4,2). (2) Passing through point (3,4), touching the x-axis, with the center on the line y=x-1.'
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'Problem to find the coordinates of point P. Find the coordinates of point P (x, y) lying on the line connecting two points A (6, -3) and B (1, 7).'
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'Outside of the circle (x-2)^{2}+(y-1)^{2}=5. Including the boundary.'
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'The coordinates of the centroid G of triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3) are (\\frac{x1+x2+x3}{3}, \\frac{y1+y2+y3}{3})'
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'Maths II river 36 books p.119\n(1) The radius r is the distance between the center (-5,4) and the origin, so r^2=(-5)^2+4^2=41\nTherefore, the equation of the circle we seek is (x+5)^2+(y-4)^2=41\n(2) The center is the midpoint of the diameter, so its coordinates are (-3+3)/2, (6+(-2))/2 which is (0,2)\nThe radius r is the distance between the center (0,2) and the point A(-3,6), so r^2=(-3-0)^2+(6-2)^2=25\nTherefore, the equation of the circle we seek is x^2+(y-2)^2=25\nAnother solution (2) On the circumference, let P(x, y) be a point different from A, B, then AP ⊥ BP so, when x ≠ -3, x ≠ 3, (y-6) / (x-(-3)) * (y-(-2)) / (x-3) = -1\nTherefore, (x+3)(x-3)+(y-6)(y+2)=0 which is x^2+(y-2)^2=25\nThis equation holds when x=-3, x=3, i.e., the points (-3,6), (-3,-2), (3,6), (3,-2) satisfy it, so this is the equation of the circle we seek.'
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'For a standard angle, determine whether the following proposition is true and explain why.\n"There are no angles greater than 360 degrees"'
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'(5) A line parallel to the y-axis is perpendicular to the x-axis. Since the x-coordinate of the point it passes through is 5, we have x=5'
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'If a parabola and a circle are tangent at one point, as shown in the figure, if the parabola and circle are tangent at the point (0,3) or the point (0,-3), then a = ±3, so the desired value of a is a = -37/4, ±3.'
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'Important Example 58: Intersection Points of Parabola and Circle\nLet r be a positive constant. Consider the parabola y=x^{2} and circle x^{2}+(y-2)^{2}=r^{2}, and answer the following questions:\n(1) When r=2, find all coordinates of intersection points between the parabola and the circle.\n(2) Investigate how the number of intersection points between the parabola and the circle changes as r varies over all positive real values.'
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'Find the values of a when the lines (a-2)x+ay+2=0 and x+(a-2)y+1=0 are parallel, coincident, or perpendicular.'
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'Understand the formula for the distance between 2 points in a plane. Find the formula for the distance between points O(0; 0), A(x_{1}, y_{1}), B(x_{2}, y_{2}).'
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'For the two circles \ x^{2}+y^{2}-2 x-4 y+1=0, x^{2}+y^{2}=5 \:\n(1) Find the equation of the line passing through the two intersection points of the two circles.\n(2) Find the center and radius of the circle passing through the two intersection points of the two circles and the point (1,3).'
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'Which of the following lines are parallel to each other, and which are perpendicular?\n(1) y=2x+3\n(2) y=√2x-1\n(3) y=-2x+1\n(4) 2x-√2y+1=0\n(5) x+2y-5=0'
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'(1) Find the equation of a circle with center (-5,4) passing through the origin.\n(2) Find the equation of a circle with diameter AB where A(-3,6) and B(3,-2).'
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'Given circles, denoted as C_1 and C_2, respectively. (1) Let the coordinates of the point of tangency on circle C_1 be (x_1, y_1) such that x_1^2 + y_1^2 = 9'
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'Important Example 49 Distance between a point on a parabola and a line\nGiven two points A(0,1) and B(2,5) and the parabola y=x^{2}+4x+7. Let there be a moving point P on the parabola.\n\nFind the minimum value of the area S of triangle PAB.'
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'Plot the range of existence of point (a, b) on the ab plane when the line segment connecting points A(1,-2) and B(-2,1) intersects the parabola y=x^{2}+ax+b at only one point other than A and B.'
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'Calculate the distance between two points on a plane.'
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'The equation of the line AB is x/a + y/b = 1. Let point P(a, b) and the distance between point P and the line AB be d. Find the maximum value of d.'
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'Let the equation of the required circle be (x-1)^(2)+(y+√3)^(2)=r^(2) (r>0). The condition for the circle (2) to be tangent to circle C is 0<r<5 and √((1-0)^(2)+(-√3-0)^(2))=5-r, therefore r=5-√4=3. Hence, the required equation is (x-1)^(2)+(y+√3)^(2)=9'
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'Find the coordinates of point Q:\n\nLet the coordinates of point Q be (x, y).\n(1) Let OP=r, and let the angle between OP and the positive direction of the x-axis be α, then r cosα=-2, r sinα=3.\nTherefore, x=r cos(α+5/6π)=r cosα cos5/6π-r sinα sin5/6π.'
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'Coordinates of points\nLet the points A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) be given.\nFind the distance between two points.\nAB=√((x₂-x₁)² + (y₂-y₁)²)\nIn particular, the distance between the origin O and A is OA=√(x₁² + y₁²)'
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'Example 55 Tangent Conditions and Equations of Circles and Lines'
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'Taking point B as the origin and edge BC as the x-axis, the coordinates of each vertex can be represented as A(0, a), B(0, 0), C(b, 0), D(b, a). Prove that PA² + PC² = PB² + PD².'
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'On a plane, there are n circles such that any two circles intersect each other and no three or more circles intersect at the same point. How many parts does the plane get divided into by these circles?'
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'Similar question Taking a point P(1/2, 1/4) on the coordinate plane. When two points Q(α, α^2) and R(β, β^2) on the parabola y=x^2 move such that the three points P, Q, R form an isosceles triangle with QR as the base, find the locus of the centroid G(X, Y) of triangle PQR. [University of Tokyo]'
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'The parabola and the circle have 4 shared points when the vertex of the parabola is on the line segment connecting point (0, -37/4) and point (0, -3) (excluding the endpoints) as shown in the figure. Therefore, -37/4 < a < -3.'
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'Two circles tangent to the coordinate axes and a line'
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'Taking the line BC as the x-axis and the perpendicular bisector of side BC as the y-axis, the midpoint L of side BC becomes the origin O, and the coordinates of each vertex can be represented as A(a, b), B(-c, 0), C(c, 0). In this case, L(0,0), M((a+c)/2, b/2), N((a-c)/2, b/2), therefore, the coordinates of the intersection points dividing the three medians AL, BM, CN in a 2:1 ratio are ((a/3), (b/3)), ((-c+(a+c))/(2+1), (0+b)/(2+1)), ((c+(a-c))/(2+1), (0+b)/(2+1)), all of which are ((a/3), (b/3)), so the three medians AL, BM, CN intersect at this point.'
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'Practice 1: Find the radius and total area of the incircles of equilateral triangles'
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'Illustrate the radius of the following angles. Also, identify the quadrant in which they lie.'
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'(1)\n{% raw %}\\(\\mathrm{AB}^{2}=(0-4)^{2}+(2-0)^{2}=20\\)\\(\\mathrm{BC}^{2}=(3-0)^{2}+(3-2)^{2}=10\\)\\(\\mathrm{CA}^{2}=(4-3)^{2}+(0-3)^{2}=10\\)\\{% endraw %}\nTherefore, BC=CA, BC^2 + CA^2 = AB^2, so ΔABC is a right-angled isosceles triangle with ∠C=90∘.'
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'Practice Let real number t satisfy 0<t<1, consider the 4 points O(0,0), A(0,1), B(1,0), C(t,0) on the coordinate plane. Also, define point D on segment AB such that ∠ACO=∠BCD. Find the maximum area of triangle ACD. [University of Tokyo]'
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'When the point (x, y) moves inside a circle with radius 1 centered at the origin, depict the range of motion of the point (x+y, x y).'
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'For the circle , answer the following questions:\n1. Find the equation of a circle with center at that is tangent to circle externally.\n2. Find the equation of a circle with center at that is tangent to circle internally.'
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'Find the arc length and area of a sector with radius 4 and central angle 150 degrees.'
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'Let a and b be positive real numbers. The parabolas C1: y = x^2 - a and C2: y = -b(x - 2)^2 are both tangent to the line ℓ at the point P(x0, y0). Define S1 as the area enclosed by the line x = 0, the parabola C1, and the tangent line ℓ, and S2 as the area enclosed by the line x = 2, the parabola C2, and the tangent line ℓ. Answer the following questions:\n(1) Express a, x0, y0.\n(2) Express the ratio of areas S1 : S2 in terms of b.'
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'Represent the set of points (x, y) that satisfy y=x+1 in a figure with a straight line as the boundary. Also, represent the regions of points that satisfy y>x+1 and y<x+1 in a figure.'
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'Find the coordinates of a point P on the x-axis equidistant from points A(-1,2) and B(3,4).'
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'Find the equation of a circle passing through the point (2,1) and tangent to the x-axis and y-axis.'
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'Basic Example 70: Given A(-2,1), B(6,-3), C(1,7), find the coordinates of the following points.'
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'Investigate how the number of intersection points between the circle (x-1)^2+(y-1)^2=r^2 and the line y=2x-3 changes depending on the radius r.'
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'Given the three vertices A(5,-2), B(1,5), C(-1,2), find the lengths of the three sides of triangle ABC and determine what type of triangle it is.'
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'Plot the region represented by the following inequalities.'
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'Find the equation of the tangent line drawn from point A(3,1) to the circle x^2+y^2=2 and the coordinates of the point of tangency.'
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'Let A and B be the intersection points of the parabola y=9-x^{2} and the x-axis. When a trapezoid is inscribed in the area enclosed by this parabola and the x-axis, with segment AB as the base, determine the maximum area of this trapezoid.'
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'Find the number of lattice points within the area enclosed by y = -x^2 + 8x and y = x (including the boundary).'
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"A circle with center C(a, b) and a constant distance r(>0) from C is a collection of points with C as the center and radius r. The circle with center C is simply called circle C, and the equation satisfied by any point (x, y) on the circle is called its equation. Let's try to find the equation of this circle. The condition for a point P(x, y) to be on circle C is CP = r, expressed in coordinates as √((x-a)^2 + (y-b)^2) = r, squaring both sides gives (x-a)^2 + (y-b)^2 = r^2. Since both sides of (2) are positive, (1)⇔(2)⇔(3), hence (3) is the equation of the desired circle. The form of equation (3) with knowledge of the center (a, b) and radius r is called the basic form of the circle equation. The equation of a circle with radius r and center (a, b) is (x-a)^2 + (y-b)^2 = r^2. The equation of a circle with radius r and center at the origin is x^2 + y^2 = r^2. Note that setting a=b=0 in 1 results in 2. When r=1, it is called a unit circle. Additionally, 1 can be considered as a translation of 2 parallel to a along the x-axis and b along the y-axis."
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'(1) Let the circle (x-1)^2 + (y+2)^2 = 9 be C. When the circle (x+1)^2 + (y-1)^2 = 4 is C₁, determine the position relationship between C and C₁.'
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'Point D is in the fourth quadrant, Circle D is tangent to the x-axis and y-axis, so the coordinates of point D can be assumed as (d, -d) and the radius is d. Since point D is below the line l, we have 3d - 4d - 12 < 0. The distance between point D and line l is |3d - 4d - 12| /√(3^2+4^2) = (d + 12) / 5. Since Circle D is tangent to line l, the distance between point D and line l is (d + 12) / 5 = d. Therefore, d = 3.'
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'Find the equation of the tangent line at point A on the circle with A(0,3) and B(8,9) as the diameter.'
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'Find the value of a when the two circles touch each other.'
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'Find the equation of a line that is tangent to the circle x^2 + y^2 = 9 and parallel to the line 4x + 3y - 5 = 0.'
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'For points A(0,1) and B(4,-1): (1) Find the equation of a circle C1 with center on the line y=x-1 passing through points A and B. (2) Find the equation of a circle C2 which is symmetric to the circle C1 found in (1) with respect to the line AB. (3) Let P and Q be points on circles C1 and C2, respectively. Find the maximum length of the line segment PQ. [Gunma University]'
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'There is a parallelogram ABCD with vertices A(-2,3), B(5,4), and C(3,-1). Find the coordinates of vertex D and the intersection point P of the diagonals.'
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'Find the equation of the following circle:\n(1) A circle with center at (1,1) that is tangent to the line 2x-y-11=0'
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'Given points A(6,0) and B(3,3), when point P moves on the circle x^2+y^2=9, find the locus of the centroid G of triangle ABP.'
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"Let C and C' be the two intersection points A, B, and the midpoint of line segment AB be M. Therefore, the length of segment OM is equal to the distance between the origin O and the line ℓ."
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'Using the coordinates (p, q) of point B, determine the condition for the line AB to be perpendicular to the line ℓ when the slope of ℓ is 2.'
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'When exactly 2 tangent lines can be drawn from point P(1, ) to curve C, answer the following questions. (i) Find the equations of the 2 tangent lines. (ii) Let Q and R be the points of tangency between the lines found in (i) and curve C. Assume that the x-coordinate of Q is less than the x-coordinate of R. Find the area S of the figure enclosed by line segment PQ, line segment PR, and curve C.'
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'Investigate the positional relationship between the following circles and lines, and find the coordinates of the intersection points if they exist.'
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'What kind of shapes do the following equations represent?'
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'Assume there are four circles on the coordinate plane that touch the x-axis, y-axis, and the line 3x + 4y - 12 = 0. Arrange the radii of these circles in ascending order and explain the relationship between the center of each circle and the line.'
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'Find the slope of the line that makes an angle of \\frac{\\pi}{4} with the line x - \\sqrt{3} y = 0.'
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'For the point A(-2, -3), find the coordinates of point Q which is symmetrical to point P(3, 7).'
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'Find the equation of the tangent line at point P on the following circle.'
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'Let A be the intersection of the two lines \ 3 x+2 y-4=0 \ (1) and \ x+y+2=0 \ (2). Determine the equation of the line passing through point A and B(3,-2) for (1). Determine the equation of the line passing through point A and parallel to the line \ x-2 y+3=0 \ for (2).'
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'Given A(-2,-3), B(3,7), C(5,2), find the coordinates of the following points.'
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"Let's find the equation of the tangent line at point (a, b) on the circle x^2 + y^2 = r^2."
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'Let line 3x+2y-4=0 be (1) and x+y+2=0 be (2), with A as the intersection point of the two lines. Find the equation of the line passing through A and point B(3,-2).'
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'Find the equation of the line that intersects the x-axis at (2, 0) and the y-axis at (0, -3).'
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'What is the shape of the triangle ABC formed by the following 3 points?'
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'Find the equations of the following circles:\n1. Circle with center at (2, -3) and radius 1\n2. Circle with center at (3, 4) passing through the origin\n3. Circle with diameter defined by the points (3, 1) and (-5, 7)\n4. Circle with center at (5, 2) tangent to the y-axis'
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'Find the center and radius of the circle that passes through the two intersection points of the two circles \\( x^{2}+y^{2}=2,(x-1)^{2}+(y+1)^{2}=1 \\) and is tangent to the line \ y=x \.'
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'Point A is in the first quadrant, Circle A is tangent to the x-axis and y-axis, so the coordinates of point A can be assumed as (a, a), with the radius being a. Since point A is below line l, we have 3a + 4a - 12 < 0. The distance between point A and line l is |3a + 4a - 12| / √(3^2 + 4^2) = (-7a + 12) / 5. Since circle A is tangent to line l, the distance between point A and line l is a, hence (-7a + 12) / 5 = a, leading to a = 1.'
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'Master the formulas for internal and external division points coordinates and conquer example 74!'
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'For which values of the constant k does the circle C: x^2+y^2+(k-2)x-ky+2k-16=0 pass through the points A(x, y) and B(x, y)? Here, . The line segment AB will be a diameter of the circle C only when k=.'
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'Show the region boundaries represented by the inequalities.'
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'Problem to find the position relationship between a line and a circle, along with the coordinates of their intersection points.'
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"The equation of the line passing through the two intersection points C and C' is \\square x+\\square y=15. Also, the area S of the triangle with the two intersection points and the origin O as vertices is S=\\square."
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'What kind of shapes do these equations represent?'
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'Are the following two lines parallel or perpendicular?'
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'Master the formula for distance between a point and a line, conquer example 83!'
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'Illustrate the radius of angle θ and specify in which quadrant the angle lies'
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'When the triangle ABC is a right triangle with vertices A(1,1), B(2,4), and C(a,0), find the value of the constant a.'
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'Find the coordinates of the point P on the circumference of the circle with equation x^2-2x+y^2-4y+4=0 that is closest to the point A(-1,1). Also, find the distance between the points A and P.'
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'(2) Right isosceles triangle where \ \\angle \\mathrm{A}=90^{\\circ} \'
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'Problem of finding the distance between two points on a plane.'
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'Given the circle TR: x^{2}+y^{2}=1, denoted as C_{0}, and let C_{1} be the circle obtained by translating C_{0} 2a units in the positive direction of the x-axis, where a is 0<a<1. Also, let A and B be the two intersection points of C_{0} and C_{1} in the first quadrant, and let P(s, t) be a point on C_{0} different from points A and B. Find the locus of the centroid G of triangle PAB as P moves on the part of C_{0} excluding the two points A and B.'
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"The position of point P on a plane is represented by a pair of real numbers, for example, (a, b). This pair (a, b) is called the coordinates of point P, where a is the x-coordinate and b is the y-coordinate. The point P with coordinates (a, b) is denoted as P(a, b). In this section, let's learn about points on a plane. The plane with coordinates is divided into 4 parts by coordinate axes. These parts are called quadrants, and they are named as the first quadrant, the second quadrant, the third quadrant, and the fourth quadrant in a counterclockwise manner. Note that the coordinate axes are not included in any quadrant. In the diagram, (+, +) indicates the signs of x and y coordinates in each quadrant."
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'In this case, the distance between the center (0,0) of circle (1) and the line (2) is equal to the radius of the circle √k, so'
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'Find the number of tangents that can be drawn from point P(1, ) to the curve C: y=x^3-x.'
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'Given the center and radius of a circle, find the equation of the circle.'
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'Find the equation of a circle with center (a, b) and radius r.'
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'Find the circle passing through the intersection of 2 circles'
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'Prove that for an acute triangle ABC, the following equation holds: tan A + tan B + tan C = tan A tan B tan C.'
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'Find the locus of the midpoint P of the line segment connecting point A(2,0) and point Q as point Q moves along the circle x^2 + y^2 = 1.'
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'Examine the position relationships between the following circles and lines, and if there are common points, find their coordinates.'
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'Since the center of circle C3 is the origin O, the distance between circle C and circle C3 is PO=√(1^2+(-2)^2)=√5\nLet r3 be the radius of circle C3, as circle C3 is inscribed in circle C, we have r3 < 3 and √5 = 3 - r3\nTherefore r3 = 3 - √5\nHence, the equation of circle C3 is x^2 + y^2 = (3 - √5)^2'
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'Plot the regions represented by the following inequalities.'
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'There is a parallelogram ABCD with vertices A(-2,3), B(5,4), and C(3,-1). Find the coordinates of vertex D and the intersection point P of the diagonals.'
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'Find the coordinates of the midpoint and the length of the segment cut by the circle with center (2, 1) and radius 2 from the line y=-2x+3.'
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'Find the equation of the tangent line drawn from point A(7,1) to the circle x^2+y^2=25.'
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'Find the equation of a circle passing through the points (0,2) and (-1,1) with its center on the line y=2x-8.'
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'When a parabola (1) and a circle (2) have 4 common points, find the range of r.'
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'Find the center and radius of the circle passing through the two intersection points of the two circles x^2+y^2=2 and (x-1)^2+(y+1)^2=1, which is tangent to the line y=x.'
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'In Figure 6, what is the length that can be read with the vernier calipers in increments of how many millimeters?'
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'What are the values on the vertical axis for points (2) and (3)?'
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'2021 Shibuya Academy Makuhari Middle School (1st time) (4)\nAs shown in Figure 5-1, there is a rectangular prism with a rhombus base and all rectangular side faces. Points are located on the edges respectively, with , .\nAdditionally, point O is located on the diagonal of the rhombus , with .\nConnecting each vertex of quadrilateral to point O creates pyramid O-KLMN. Answer the following questions. The volume of the pyramid can be calculated as (base area) x (height ÷ 3).'
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'Explain the difference between a bright red star and a dark red star.'
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'There is a right triangle ABC as shown in the figure 2, and squares with sides of AD, BD, and CD respectively. In this case, what is the area of the square with CD as one side in square centimeters?'
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'What is the size of cell A in figure 12 (length between PQ) in micrometers? Use the value obtained in (5) and answer in integers.'
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'(3) As shown in the graph on the right, cliff B is 48m above sea level and at a distance of 70m north from point A, cliff C is 53m above sea level and at a distance of 70m south from point A, simply note down their respective positions.'
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'(2) The line drawn by point O becomes like a bold line. Firstly, the central angle of a semi-circle with a radius of 6 cm is between (2) and (3). Adding the part between 8 and 9 (which is equal to the length of a 60-degree arc) gives a total of 180×3+90+60×2 = 750 degrees. Moreover, the arc between 3 and 4 has a radius of 12+6=18 cm and a central angle of 30 degrees. Therefore, the length of the line drawn by point O is calculated as 6×2×3.14×750/360+18×2×3.14×30/360=(25+3)×3.14=87.92 cm.'
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'In the figure 5-1, there is a right-angled triangle with angle A, where AB=3 cm and AC=6 cm, and a right-angled isosceles triangle with angle D, where DE and DF are both 6 cm. Answer the following questions about the geometric shape formed by combining these right-angled triangles. Take the value of Pi as 3.14. Also, note that the volume of a cone can be calculated by base area * height / 3.'
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'The height above sea level of volcanic ash layer X is 53 meters at cliff A and 44 meters at cliff B. When marked with circles on a graph, it looks as shown in the right graph.'
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'(4) A set of points where the difference in distance from A to the sound source and B to the sound source is constant at 350m forms a line, indicating the presence of the sound source. This is represented by (I). A curve where the difference in distance from two points to the sound source is constant is called a hyperbola.'
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"Comets are celestial bodies in the solar system that revolve around the sun like planets. Comets exhibit a distinctive feature of suddenly brightening as they approach the sun from far away in the solar system and darkening abruptly and disappearing as they move away. Additionally, as shown in Figure 6, comets present a different appearance with a long tail waving unlike other celestial bodies. The tail of a comet extends in the opposite direction of the sun. (5) Suppose a new comet is discovered, and it is visible immediately after the sunset of that day. Describe the appearance of the comet's tail as a straight line."
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'(3) The sound source A is at the position reached within 1 second, and B is at the position reached within 2 seconds. Therefore, drawing a circle with a radius of 350 × 1 = 350 meters centered at A, and a circle with a radius of 350 × 2 = 700 meters centered at B, the two intersection points of these two circles will indicate the position of the sound source.'
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'Choose one correct statement that can be inferred from the graph in Figure 4.'
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'Find the length of each side of the following triangle.'
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'Equation of a plane perpendicular to two coordinate axes'
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'Let the polar coordinates of point A be (r₁, θ₁) and those of point B be (r₂, θ₂). Find the area of triangle OAB, denoted by S.'
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'In parallelogram ABCD, let M be the midpoint of side AB, E be the point that divides side BC into 1:2, and F be the point that divides side CD into 3:1. If →AB=b and →AD=d'
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'A regular hexagon ABCDEF with side length 1 is given. When point P moves along side AB and point Q moves along side CD independently, find the area where point R, which divides segment PQ in the ratio 2:1, can pass through.'
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'Although it is possible to directly substitute z=x+yi into the equation (3)(2) and calculate it, the computation becomes very complicated (see the first consideration after the answer). Therefore, we first consider the equation of an ellipse with congruence to shape C and having the focus on the x-axis, then rotate it to find the equation of C. (1) K: \\frac{x^{2}}{2^{2}}+\\frac{y^{2}}{1^{2}}=1 The coordinates of the foci are, \\sqrt{2^{2}-1^{2}}=\\sqrt{3}, so it is (\\sqrt{3}, 0),(-\\sqrt{3}, 0). The length of the major axis is 2\\cdot2 and the length of the minor axis is 2\\cdot1, therefore, the area to find is \\pi\\cdot2\\cdot1=2\\pi.'
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'The tangent at any point on the curve C in the first quadrant always intersects the positive parts of the x-axis and y-axis, and the intersection points are denoted as Q and R, respectively. The point P divides the segment QR internally in the ratio 2:1.'
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'For polar coordinates, find the equations of the following circle and line: (1) A circle with center at point A(3, π/3) and radius 2. (2) A line passing through point A(2, π/4) and perpendicular to OA (O is the pole).'
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'(2) 128\nIn the isosceles trapezoid \ \\mathrm{ABCD} \ where \ \\mathrm{AB}=2 \\mathrm{~cm}, \\mathrm{BC}=4 \\mathrm{~cm}, \\angle \\mathrm{B}=60^{\\circ} \, when \ \\angle \\mathrm{B} \ increases by \ 1^{\\circ} \, by how much does the area \ S \ of the trapezoid \ \\mathrm{ABCD} \ increase? Assume \ \\pi=3.14 \.'
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'Find the position vector of the orthocenter of a triangle.'
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'In the coordinate space, let A(1,0,2) and B(0,1,1). When point P moves along the x-axis, find the minimum value of AP+PB.'
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'On the coordinate plane, the circle C passes through the point (0,0), its center lies on the line x+y=0, and it is tangent to the hyperbola xy=1. Find the equation of circle C. Here, the circles and hyperbolas are said to touch at a point if the tangents of the circle and the hyperbola coincide at that point.'
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'For the ellipse , find the locus of the point outside the ellipse from which two tangents drawn to the ellipse intersect at right angle.\n[Type University of Tokyo]\nBasic 155'
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'Find the geometric figure represented by all points P(z) that satisfy the equation .'
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'On the curve \\sqrt[3]{x}+\\sqrt[3]{y}=1, let \\mathrm{P} be the point in the first quadrant where the tangent intersects the x-axis and y-axis at points \\mathrm{A}, \\mathrm{B} respectively. If the origin is \\mathrm{O}, find the minimum value of \\mathrm{OA}+\\mathrm{OB}.'
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'Find the coordinates and length of the chord formed by the intersection of the following line and curve.'
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'Basic Concepts 1 Polar and Straight-line Equations (1) Polar equation of a circle with center at the pole O and radius a r=a r=2a cos θ r^2-2r r₀ cos(θ-θ₀)+r₀^2=a^2 θ=α r cos (θ-α)=a (a>0) (2) Circle with center at (a, 0) and radius a r=2a cos θ (3) Circle with center at (r₀, θ₀) and radius a r^2-2r r₀ cos(θ-θ₀)+r₀^2=a^2 (4) Line passing through the pole O and forming an angle α with the initial line θ=α (5) Line passing through point A(a, α) and perpendicular to OA'
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'Let the equation represent a conic section . Also, let be constants, and the equation represent a hyperbola . Determine the values of when the two foci of and the two foci of form the 4 vertices of a square.'
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'Using the 4 points given in question (1) A(2,4), B(-3,2), C(-1,-7), D(4,-5), form a quadrilateral\n(2) Using the 3 points A(0,2), B(-1,-1), C(3,0) as vertices, connect another point D to form a parallelogram. Find the coordinates of the fourth vertex D.'
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'(4) For the plane PQR and edge OD, the situations are as follows. When q = 1/4, plane PQR is □. When q = 1/5, plane PQR is 又. When q = 1/6, plane PQR is ネ. Choose the one that fits two 〜 ネ, one from 0 to 5, you can choose the same option repeatedly.'
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'Find the locus of the center P of a circle that is tangent to both the circle and the line .'
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'For the ellipse \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0) \\), the coordinates of the foci are \\(\\left(\\sqrt{a^{2}-b^{2}}, 0\\right),\\left(-\\sqrt{a^{2}-b^{2}}, 0\\right)\\). The foci are on the x-axis, with the length of the major axis being 2a and the length of the minor axis being 2b.'
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'Prove that the difference in distance from any point on the hyperbola to its two foci is constant.'
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'State the condition for three distinct points A(α), B(β), C(γ) to be collinear.'
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'Let the two endpoints A and B of a line segment of length 2 move along the x-axis and y-axis, respectively. When , find the trajectory of point P.'
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'Passing through the point A(3, -4), find the line parallel to the line l: 2x-3y+6=0 and name it g. Determine the equation of the line g.'
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'In the regular hexagon ABCDEF, with center O, point P divides side CD internally in the ratio 2:1, and point Q is the midpoint of side EF. If vector AB is a and vector AF is b, express vectors BC, EF, CE, AC, BD, QP in terms of vectors a and b.'
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'When a point P(z) moves along the perimeter of a circle centered at -i with a radius of 1 (excluding the origin), the point Q(w) represented by (3) 114 w=1/z, what kind of shape does it trace?'
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"Topic: Investigation of quadratics represented by complex number equations and rotation movement Mathematics In Chapter 3 of mathematics C, we learned about geometric shapes in the complex plane, and in Chapter 4 we learned about the properties of quadratics. Here, we will investigate cases where the shape represented by the complex number z's equation is a quadratic. First, let's confirm the basic concepts of quadratics with the following problem. CHECK 3-A The equation of the trajectory of point P with a total distance of 6 from points F(√5, 0) and F'(-√5, 0) is to be found."
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'Plot the region E on the ab-plane consisting of all points (a, b) where the hyperbola and the line have common points.'
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'Assume that △ABC is an equilateral triangle with vertices A(-1), B(1), and C(√3 i). Prove that when △PQR with vertices P(α), Q(β), R(γ) is also an equilateral triangle, the equation α² + β² + γ² - αβ - βγ - γα = 0 holds true.'
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'Prove the equation of the tangent line at the point (x0, y0) on the circle using vectors.'
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'In triangle △OAB with vertices O(0), A(1), B(ι), where ∠O is the right angle vertex, prove that when triangle △PQR is formed by points P(α), Q(β), R(γ) with ∠P as the right angle vertex, the equation 2α² + β² + γ² - 2αβ - 2αγ = 0 holds.'
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'(2) The sum of the distances from point z to the 2 points (√3+3i)/2 and -(√3+3i)/2 is constant at 4, therefore, the figure C is an ellipse with the 2 foci at (√3+3i)/2 and -(√3+3i)/2. Let c be the distance from the origin, which is the center of this ellipse, to the foci. The coordinates of the foci on the xy-plane are (c, 0) and (-c, 0). This ellipse is congruent to an ellipse where the sum of the distances from points on the ellipse to the 2 foci is also 4.'
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'Consider a circle C with radius a in the first quadrant of the xy-plane, which is tangent to both the line l: y=mx(m>0) and the x-axis. Also, consider circles tangent to the line l, the x-axis, and the circle C at one point each with a radius of b, where b>a. (1) Express t in terms of m. (2) Express b/a in terms of t. (3) Find the limit lim_{m \to +0} 1/m(b/a-1).'
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'(1) Let the lengths of the three sides of triangle ABC be AB=8, BC=7, CA=9. Let vector AB=b and vector AC=c, and let P be the incenter of triangle ABC. Express vector AP in terms of b and c.'
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'Midpoint theorem: In triangle ABC, let the midpoints of sides AB and AC be M and N respectively. Then MN // BC and MN = 1/2 BC'
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'There is a quadrilateral ABCD inscribed in a circle. When AB=4, BC=5, CD=7, DA=10, find the area S of quadrilateral ABCD.'
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'Determine the range of values for x so that a triangle with side lengths 3, 5, and x becomes an acute triangle.'
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'A regular tetrahedron OABC with edge length of 6 is given. Let L be the midpoint of edge OA, M be the point that divides edge OB into 2:1, and N be the point that divides edge OC into 1:2. Find the area of triangle LMN.'
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'In triangle ABC, if B=30°, b=√2, and c=2, find the values of A, C, and a.'
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'Please provide the terms related to the circumcircle of a triangle and their definitions.'
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'The center of the circle passing through points A, P, and Q is the point of intersection of the perpendicular bisectors of two chords.'
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'Using the power of a point theorem, show the properties of the two tangents drawn from point P to a circle.'
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'For a regular octagon, find the following numbers.\n(1) The number of quadrilaterals that can be formed by connecting 4 vertices\n(2) The number of triangles formed by connecting 3 vertices that share an edge with the regular octagon'
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'On the line x=1, the point T is located where the y-coordinate is √3. The point P is the intersection of the line OT and a semicircle with radius 1. The angle we are looking for is ∠AOP.'
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'On a rectangular floor with dimensions of 240 cm by 396 cm, we want to cover it with square tiles of side length a cm without any gaps. Find the maximum value of a in this case. Also, determine the number of tiles that can be laid out.'
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'Explain and prove the properties of the circumcenter, incenter, and centroid of a triangle.'
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'Determine whether the four points A, B, C, D on the right diagram lie on the same circle.'
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'The radius of circumcircle is \ \\frac{85}{8} \, and the radius of incircle is 2'
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'In a quadrilateral ABCD inscribed in a circle, with AB = 8, BC = 10, and CD = DA = 3. Find the area S of the quadrilateral ABCD.'
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'79. In triangle ABC, AB=2, BC=4, CA=2√3. Let AD be the altitude from vertex A to side BC, and let E and F be the points of intersection of the circle with diameter AD and sides AB and CA, respectively. E and F are different from A. [Tokyo Jikeikai Medical University]\n(1) Prove that points E, B, C, and F lie on the same circle.\n(2) Find the area of triangle EBF.'
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'As shown in the diagram, label all vertices of an equilateral triangle with side length 2 and each midpoint of the edges from 1 to 6. Match the outcome of the first dice roll with this number. Connect the points corresponding to the numbers rolled on the dice three times to create a shape. Find the expected value of the area of the resulting shape.'
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'In \ \\triangle ABC \, with \ \\angle A=90^{\\circ}, \\angle B=60^{\\circ}, \\angle C=30^{\\circ} \ and noting that \ AD \ is the diameter of a circle, draw the auxiliary lines \ AD, ED, EF, DF \.'
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'Let p and q be the lengths of the diagonals AC and BD of quadrilateral ABCD, and let θ be one of the angles formed by the diagonals. Express the area S of quadrilateral ABCD in terms of p, q, and θ.'
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'Please explain about points inside and outside a circle and the sizes of angles.'
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'In triangle ABC, O is the circumcenter. Find the angles α and β in the figure on the right.'
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'Conditions for a parallelogram: A quadrilateral is a parallelogram if any of the following conditions are met. [1] Two pairs of opposite sides are parallel. [2] Two pairs of opposite sides are equal. [3] Two pairs of opposite angles are equal. [4] One pair of opposite sides is parallel and equal in length. [5] The diagonals intersect at their respective midpoints.'
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'In the quadrilateral ABCD inscribed in a circle, where AB = BC = 1, BD = √7, and DA = 2, find:\n1. The position of point A\n2. The length of side CD\n3. The area of quadrilateral ABCD S'
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'Given a quadrilateral ABCD inscribed in circle PR with AB = 4, BC = 5, CD = 7, DA = 10, find the area S of quadrilateral ABCD.'
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'Determine the minimum length of the diagonal in a rectangle with a length of 40 cm. Also, describe the shape of the rectangle at this minimum. Let the vertical length of the rectangle be x cm, then the horizontal length is (20-x) cm. Since x>0 and 20-x>0, we have 0<x<20. Denote the length of the diagonal as l cm, l^2 =x^2+(20-x)^2 =2 x^2-40 x+400 =2(x-10)^2+200 (1) where l^2 reaches the minimum value of 200 at x=10. Since l>0, when l^2 is minimized, l is also minimized. Thus, the minimum value of the diagonal length l is sqrt(200)=10 sqrt(2)(cm). At this point, the horizontal length is also 10 cm, making the rectangle a square.'
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'(2) Consider cases based on the length of one side. 11) A square formed by two vertically adjacent lines and two horizontally adjacent lines.'
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'Take point O on the plane and define two perpendicular lines at point O, as shown in the diagram to the right. These are called the x-axis and y-axis, respectively. Point O is called the origin. In this case, if point A is located at coordinates (3, 2), please provide its x-coordinate and y-coordinate.'
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'In the figure to the right, label all the vertices of an equilateral triangle of side length 2 and the midpoints of each side with numbers 1 to 6, corresponding to the rolls of a die. Roll the die 3 times and connect the numbers obtained to form a geometric shape. Find the expected value of the area of the resulting shape.'
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'There are 10 non-intersecting lines on a plane, with no three lines passing through the same point. If two of the 10 lines are parallel, determine the number of intersection points and triangles formed by these 10 lines.'
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'Example 4: Parabolic Antenna\nA parabola is called a parabola in English. The surface of a parabolic antenna used for satellite broadcasting reception is in the shape of the surface that is formed by rotating a parabola around its axis.'
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"Circles O and O' with radii 5 and 8, respectively, are externally tangent at point A. Let B and C be the points where the common external tangent of these two circles touches circles O and O'. Extend BA to intersect circle O' at point D.\n(2) Prove that points C, O', and D are collinear.\n(3) Find the ratios of AB:AC:BC."
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'Calculate the number of parallelograms formed by 3 parallel lines and 5 lines intersecting them.'
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'Distance between two points\n(1) The distance between two points A(x1, y1) and B(x2, y2) on the coordinate plane is\nAB=√((x2-x1)^2+(y2-y1)^2)\nIn particular, the distance between the origin O and point A(x1, y1) is OA=√(x1^2+y1^2)\n(2) The distance between two points A(x1, y1, z1) and B(x2, y2, z2) in coordinate space is\nAB=√((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)\nIn particular, the distance between the origin O and point A(x1, y1, z1) is OA=√(x1^2+y1^2+z1^2)'
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'For a regular decagon, find the following:\n(1) The number of diagonals\n(2) The number of triangles with 3 of the vertices of the decagon as vertices\n(3) The number of triangles from (2) that share only one side with the decagon'
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'Prove that the centroids of triangles ABC and DEF coincide.'
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'For the given line segment AB, draw the following points.'
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'Mathematics I\nTherefore, the area of triangle (ABC) is\n\nEX 1 A piece of paper in the shape of an equilateral triangle with a side length of (10 cm) is given. Let the vertices of this equilateral triangle be (A, B, C), and let point (P) on side (BC) be at a distance of (2 cm) from point (B). When folding this equilateral triangle paper so that point (A) coincides with point (P), let the intersection points of the folds with sides (AB, AC) be (D, E) respectively. At this time, if (AD=) is A(cm), (AE=) is B(cm), and the area of △ADE is C(cm^{2}).\n[From Kyoto Makie University]'
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'In triangle ABC, where BC=17, CA=10, AB=9. Find the value of sinA, the area of triangle ABC, the radius of the circumcircle, and the radius of the incircle.'
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'Basic example 85 The length of the segment cut by the parabola from the x-axis\n(1) Find the length of the segment cut by the graph of the quadratic function y=-x^{2}+3x+3 from the x-axis.\n(2) Prove that the length of the segment cut by the graph of the quadratic function y=x^{2}-2ax+a^{2}-3 from the x-axis is constant regardless of the value of the constant a.'
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'Given AB=2√2, BC=4√2, CA=2√6, find the lengths of the sides of the triangle.'
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'(2) In triangle ABC, if BC = 5, CA = 3, AB = 7. Let D and E be the points where angle A and its exterior bisector intersect line BC, respectively. Find the length of segment DE.'
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'On a plane, there are 10 lines such that no three of them intersect at the same point. When exactly 2 of the 10 lines are parallel, determine the number of intersection points formed by these 10 lines and the number of triangles formed.'
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'Plot the following points:\n(1) Point P equidistant from sides AB, BC, CA\n(2) Point Q equidistant from points A, B, C'
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'Draw the following circles based on the circle O and chord AB shown on the right. Note that points P and Q are different from A and B, and are not on the perpendicular bisector of chord AB.'
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'For a right triangle with side lengths a, b, c, where the circumradius is 3/2 and the inradius is 1/2, answer the following questions. Assume a ≥ b ≥ c:'
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'Find the minimum length of the diagonal in a rectangle with a length of 40 cm. Also, determine what kind of rectangle it will be at that time.'
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'Write the following mathematical terms and their corresponding definitions in Japanese.'
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'The length of the shorter side is at least 1 meter and at most 3 meters'
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'Find the equation of the parabola obtained by symmetrically moving the parabola y=-2x^2+3x-5 with respect to the following lines or points.'
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'Find the angle θ formed by the following two lines. Assume 0° ≤ θ ≤ 90°.(1) AB and FG (2) AE and BG (3) AF and CD'
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'In a equilateral triangle ABC with side length 1, let D divide BC in the ratio 1:2, let E divide CA in the ratio 1:2, let F divide AB in the ratio 1:2. Let P be the intersection of BE and CF, Q be the intersection of CF and AD, and R be the intersection of AD and BE. Find the area of triangle PQR.'
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'In triangle ABC, if AB = 6, BC = 7, CA = 5, find the radii of the circumcircle R and the incircle r.'
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'Please provide the terms and the pages that have shapes circumscribed by two circles.'
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'On the semicircle with a radius of 1, the point whose x-coordinate is 1/2 is the point P. The angle we are looking for is ∠AOP.'
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'Find the coordinates of the point Q, which is symmetrical to the point P(3, 4) with respect to the line y = 2x + 1.'
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'Let the lengths of the sides of triangle ABC be a, b, c. If (a+b) : (b+c) : (c+a) = 4 : 5 : 6 and the area is 15√3, then find the circumradius R and inradius r of triangle ABC.'
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'In triangle ABC, with the radius of the circumcircle denoted as R. When A=30 degrees, B=105 degrees, and a=5, find R and c.'
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'When symmetrically moved about the origin, the vertex is at the point \\( \\left(-\\frac{3}{4}, \\frac{31}{8}\\right) \\), forming a concave parabola,\n\\[ y=2\\left(x+\\frac{3}{4}\\right)^{2}+\\frac{31}{8} \\quad\\left(y=2 x^{2}+3 x+5 \\text { also valid }\\right) \\]'
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'Circle O intersects with circle B at points P and Q, and further intersects with the diameter FG of circle B at points A and center B. In addition, E is the intersection point of lines PQ and FG. If EA = x, AB = a, and BG = b, express x in terms of a and b.'
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'Find the distance between the following two points.'
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'There is an equilateral triangle paper with a side length of 10 cm. Let A, B, and C be the vertices of this equilateral triangle, and point P be a point on edge BC such that BP=2 cm. When folding this equilateral triangle paper so that vertex A coincides with point P, let the intersections of edges AB, AC, and the fold be denoted as D and E, respectively. At this point, AD= □ cm, AE= □ cm, and the area of triangle ADE is □ cm².'
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'For a triangle ABC that is not an equilateral triangle, with circumcenter O, centroid G, and orthocenter H, prove the following: (1) Let L be the midpoint of side BC, and M, N be the midpoints of segments GH and AG respectively. Prove that quadrilateral OLMN is a parallelogram. You may use the fact that AH=2OL. (2) Prove that point G lies on segment OH. (3) Prove that OG:GH=1:2.'
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'Find the length of the line segment cut from the x-axis by the graph of the following two quadratic functions:\n(A) y = 2x^2 - 8x - 15\n(B) y = x^2 - (2a + 1)x + a(a + 1) (where a is a constant)'
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'Positional relationship between a parabola and the x-axis'
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"There are two circles P and Q that intersect with two circles O and O'. As shown in the figure on the right, draw a line from point A, which is beyond P of segment QP, tangent to circle O and intersecting with circle O', with the points of tangency being C, and the points of intersection being B and D. If AB=a, BC=b, CD=c, express c in terms of a and b."
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"Prove the following using the converse of Ceva's theorem:\n1. The three medians of a triangle intersect at one point.\n2. The three angle bisectors of a triangle intersect at one point."
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'67 diagrams; in order of vertices and axes: (1) point (2, -1), line x=2 (2) point (-2, -3), line x=-2 (3) point (1, 1), line x=1'
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'Quadrilateral ABCD is inscribed in a circle O, with AB=3, BC=CD=√3, and cos ∠ABC=√3/6. Find: (1) The length of segment AC (2) The length of side AD (3) The radius R of circle O'
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'Find the area of the following shapes.\n1. Parallelogram ABCD with AB=2, BC=3, and ∠ABC=60 degrees\n2. Regular octagon circumscribed around a circle with radius 10'
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'In triangle ABC, when a=13, b=7, and c=15, find A.'
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'For a right triangle where the sum of the lengths of the two legs is 16, what shape maximizes the area? Also, calculate the maximum value.'
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"There are three cases for the position relationship between a circle and a line. Here, r is the radius of the circle, and d is the distance between the center of the circle and the line. [1] Intersection at 2 points (2 shared points) 0 ≤ d < r [2] Tangent (1 shared point) 0 ≤ d < r [3] Disjoint (no shared points) 0 ≤ d < r When there is only one shared point, the circle and the line are tangent, and this line is called the tangent, with the shared point being the point of tangency. Let's first investigate the properties of the tangents of a circle."
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'Determine the conditions for the triangles to exist'
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'In triangle ABC, where AB=6, BC=a, and CA=4, let M and N be the midpoints of BC and CA, respectively. (1) Find the value of a when AM=√10. (2) When a is the value from (1), find the length of segment BN.'
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'(2) The longest side is CA, so AB + BC = 18, CA < AB + BC, therefore triangle ABC exists.'
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"Let's review Example 65! Let's use the fact that the lengths of the two tangents drawn from a point outside the circle are equal. Once again, let's try eliminating unnecessary shapes."
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'TRAINING 112 (1)\nLet the point O on the flat square be the origin, consider the coordinate plane with the east direction as the positive direction of the x-axis and the north direction as the positive direction of the y-axis.\nPoint A is located 28 units east of point O. Additionally, point P is south of the line connecting points O and A.\nPoint P is at a distance of 25 from O and 17 from A.\n(1) Find the coordinates of point A.\n(2) Find the coordinates of point P.'
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'Practice 3: Incenter, Circumcenter, Centroid of a Triangle'
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'Solve the following problems for triangle ABC:\n(1) Find the value of cos A.\n(2) Find the area S of triangle ABC.\n(3) Find the radius r of the incircle of triangle ABC.'
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'Calculate the values of trigonometric functions in a right triangle'
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'One of the intersection points of two graphs is the point (-1,0)'
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'Prove that the points B, C, F, E lie on a single circle when a perpendicular line AD is drawn from vertex A of acute triangle ABC to side BC, and perpendicular lines DE, DF are drawn from D to sides AB and AC, respectively.'
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'In right triangle ABC, with AB > AC and ∠A = 90°, draw the perpendicular AD from vertex A to edge BC.'
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'In the pentagon ABCDE circumscribed about a circle, where AB = 7, BC = 3, CD = 5, DE = 6, ∠BCD = 120° and ∠A = 82°, find:\n(1) The length of segment BD\n(2) The length of segment AD\n(3) The length of side AE\n(4) The area of quadrilateral ABDE'
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'When A=90^{\\circ}, \\sin A=\\sin 90^{\\circ}=1, so 2R \\sin A=2R \\cdot 1=2R. Also, side BC is the diameter of the circumcircle of triangle ABC, so a=2R, hence a=2R \\sin A.'
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'(1) \\\\( \\theta=30^{\\circ}, \\\\ 150^{\\circ} \\\\\\\n(2) \\\\( \\theta=45^{\\circ} \\\\\\\n(3) \\\\( \\theta=120^{\\circ} \\\\\\\n'
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'66 diagram; (1) point (-1,0) and line x=-1 in order of vertices and axis, (2) point (1,1) and line x=1'
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'In △ABC, suppose AB = 7√3 and ∠ACB = 60°. What is the radius of the circumcircle O of △ABC? Let point P move on the arc AB containing point C of the circumcircle O.'
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'(1) Find the measures of the three angles of triangle ABC where ∠A=90°, AB=2, and BC=3.\n(2) Find the lengths of the three sides of triangle ABC where ∠A=70° and ∠B=∠C.'
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'In the diagram on the right, point I is the incenter of triangle ABC. Find the following: (1) 𝛼 (2) AI:ID'
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'Consider a coordinate plane with point O as the origin, eastward as the positive direction of the x-axis, and northward as the positive direction of the y-axis.'
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'The graph of the quadratic function y = ax^2 + 2ax + a + 6 (a≠0) intersects the x-axis at two points P and Q, and the length of line segment PQ is 2√6. Determine the value of the constant a.'
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'In the figure on the right, point O is the circumcenter of triangle ABC. Find the angles alpha and beta.'
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"When thinking as described in the question, it is unclear where to apply the properties of the learned shapes. First, point A is the point of tangency of the two circles O and O', so let's draw the common tangents of the two circles passing through point A. By focusing on a part of the figure, the relevant properties will become apparent."
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'Using the result from the previous question, find the length of a side of the next regular polygon that is inscribed in a circle with a radius of 10. Also, find the length of the perpendicular dropped from the center O of the circle to a side of the regular polygon. You may use trigonometric tables. Round the result to two decimal places.'
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'In the given figure, find the value of x. Here, PT is the tangent to the circle, and T is the point of contact.'
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'PRACTICE 4 (3) In △ABC, BC=a, CA=b, AB=c, the radius of the circumscribed circle is 3, and the area is S. In this case, S=□ABc. The answer choices are (0) 1/2 (1) 1/3 (2) 1/6 (3) 1/8 (4) 1/12'
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'In triangle ABC with AB=6, BC=a, and CA=4, let M and N be the midpoints of sides BC and CA, respectively. (1) Find the value of a when AM=√10. (2) When a has the value from (1), find the length of segment BN.'
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'(1) In triangle ABC, if a=1, b=√3, and A=30°, find the lengths of the remaining side and the size of the angle.'
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'Example 1 In the square ABCD with side length 8, points P, Q, and R are taken on the sides AB, BC, and CD respectively such that AP=x, BQ=2x, CR=x+4(0<x<4). Since the areas of triangles PBQ, QCR are represented by A, B respectively when expressed in terms of x, the area of triangle PQR is minimized when x=.'
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'Find the area of the quadrilateral ABCD, where 77^{3} AB =5, BC=6, CD=5, DA=3, and ∠ADC=120^{\\circ}.'
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'Inside angle XOY, there is a point A such that ∠XOA=30° and OA=3. When points P and Q are taken on OX and OY respectively, find the minimum value of AP+PQ+QA.'
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'On a rectangular floor of 2m 40cm by 3m 72cm, you want to lay square tiles of side length a cm without any gaps. Find the maximum value of a. Also, determine the number of tiles that can be laid.'
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'Explain problems related to the sides and angles of a triangle, and prove the following theorems:\n1. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.\n2. The difference between the lengths of any two sides of a triangle is less than the length of the third side.'
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"Prove that when a circle O passing through the common chord AB of two intersecting circles O and O', and chord CD of circle O and chord EF of circle O', the four points C, D, E, F are concyclic. It is given that the four points C, D, E, F are not collinear."
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'In △ABC, where the radius of the circumcircle is R. Find the following: (1) when a=10, A=30°, B=45°, find C, b, R (2) when b=3, B=60°, C=75°, find A, a, R (3) when c=2, R=√2, find C'
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'74 degrees isosceles triangle, the maximum value is 32'
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'(1) Which of the following quadrilaterals is inscribed in a circle?\n(2) In acute triangle ABC, point D is taken on side BC (different from B and C), and perpendiculars DE and DF are drawn from point D to sides AB and AC respectively. Prove that quadrilateral AEDF is inscribed in a circle.'
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'In a town where the roads are like a grid, find the shortest path from point A to point B.\n(1) How many possible routes are there?\n(2) How many of the routes pass through point C from (1)?\n(3) How many of the routes from (1) do not pass through point C?'
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'Explanation using the condition of a quadrilateral inscribed in a circle\n(1) Which of the right quadrilaterals ABCD can be inscribed in a circle?\n(2) There is a quadrilateral ABCD inscribed in a circle, and a line parallel to side AD intersects sides AB, DC at points E, F respectively. Prove that the quadrilateral BCFE is also inscribed in the circle.'
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'Find the equations of the parabola when the given parabola in the example (1) is moved symmetrically about (1) axis (2) with respect to the origin.'
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'■Circumcenter…Intersection point of the perpendicular bisectors of the sides of a triangle\nMiddle school\nPerpendicular bisector of a line segment\nPoint P lies on the perpendicular bisector of line segment AB ⇔ PA=PB\nIn the same line\nIn other words\n“The perpendicular bisector of line segment AB is a collection of points equidistant from points A and B”'
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'(1) How many triangles can be formed by connecting 3 vertices of a regular pentagon? How many of those triangles share 2 sides with the regular pentagon? (2) How many line segments can be formed by connecting 2 vertices of a regular pentagon?'
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'Point P is on the semicircle with a radius of √5, so OP = √5\nIn the right triangle OPQ, OQ² + 2² = (√5)², hence OQ² = 1 and OQ = 1\nTherefore, the coordinates of point P are (-1,2)\n\nTherefore, sin θ = 2/√5, cos θ = -1/√5, tan θ = 2/-1 = -2'
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'When the sum of the lengths of the two sides forming a right angle triangle is 16, what shape maximizes the area of the triangle? Also determine the maximum value.'
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'In the tetrahedron ABCD with side length 4, let M be the midpoint of side CD and let the angle AMB be θ'
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'An angle ∠XOY=30° with a point A where OA=3 inside the angle. Points P and Q are taken on OX and OY respectively. Find the minimum value of AP+PQ+QA.'
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"In the figure on the right, the two circles O and O' are tangent externally. A and B are the points where the common tangent of circles O and O' intersect the circles. If the radii of circles O and O' are 6 and 4 respectively, find the length of segment AB."
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'In triangle 128 (3), when á=√6+√2, b=2, and C=45°, find the length of the remaining side and the size of the angle.'
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'There is a quadrilateral ABCD inscribed in a circle with radius TR, where the lengths of the sides are AB=√7, BC=2√7, CD=√3, and 141DA=2√3. Find: (1) the value of cos B (2) the length of the diagonal AC (3) the area S of the quadrilateral ABCD'
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'There are 4 roads running east to west and 4 roads running north to south. How many shortest paths are there for the following destinations: (1) From point A to point B. (2) From point A, passing through points C and D, to point B. (3) Of the shortest paths from point A to point B, those that pass through at least one of points C or D.'
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'A parabola y = x² and a circle x² + (y - 5/4)² = 1 touch at two different points. Find the area S of the region enclosed by the shorter arc of the circle with the two tangent points as endpoints and the parabola.'
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'(1) Distance between point A and line BC\n(2) Area of triangle ABC'
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'Given the curve y=9-x^2 intersects the x-axis at points A and B, and a trapezoid ABCD is inscribed in the region enclosed by curve and line segment AB. Find the maximum area of this trapezoid. Also, determine the coordinates of point C at that time.'
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'Find the arc length and area of the following sector:'
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'Basic Example 69 Coordinates of the centroid of a triangle'
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'On the plane, there is a parabola C: y=x^2-2 and a line l: y=4x. (1) Find the coordinates of the intersection points A and B of C and l. (2) Let moving point P on C move from A to B. Find the coordinates of point P where the area of triangle PAB is maximized.'
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'Passing through point A(3,1), find the equation of the line PQ passing through the points of tangency P and Q of two tangents that are tangent to the circle x^2 + y^2 = 5.'
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'Find the area of the triangle with vertices O(0,0), A(x1, y1), and B(x2, y2).'
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'Find the equation of a circle passing through point (4,2) and tangent to the x-axis and y-axis.'
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'Find the distance between two points A(a) and B(b) on a number line.'
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'Find the coordinates of point Q after rotating point P(4, 2√3) around the origin by π/6.'
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'Find the coordinates of the following points: (5, 1), (9, 5), (3, 9)'
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'Illustrate the region represented by the system of inequalities x² + y² - 2x + 2y - 7 ≥ 0, x ≥ y.'
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'Find the equation of the tangent to the circle x^2 + y^2 = r^2 at point (x1, y1).'
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'Let a be a constant satisfying a > 1. There is a point M(2, -1) on the coordinate plane. For a point P(s, t) different from M, point Q is taken such that the three points M, P, Q are collinear in that order, and the length of segment MQ is a times the length of segment MP.'
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'Consider a circle with radius r and center C, and a line ℓ with distance d from C. Determine the position relationship between the circle and the line based on the relationship between d and r.'
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'What will be the shape of the intersection points P(x, y) of the two lines l: tx-y=t and m: x+ty=2t+1 as t varies over real values? Find their equations and plot them.'
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'When the line (3) shares a point with region D, the slope m is maximized when the line is tangent to circle C. Calculate the maximum slope m at this point.'
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'Find the coordinates of a point equidistant from the three points A(1,5), B(0,2), and C(-1,3).'
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'What inequality does the shaded area in the figure represent? Exclude the boundary line.'
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'Consider two points A(-1,2) and B(4,2) on the coordinate plane. Let t be a real number such that 0 < t < 1. Point P divides the line segment OA in the ratio t:(1-t) and point Q divides the line segment OB in the ratio (1-t):t. Find the minimum length of the line segment PQ and the corresponding value of t.'
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'When point Q moves on the circle x^{2}+y^{2}=9, find the locus of point P that divides the line segment AQ connecting point A(1,2) and Q internally in the ratio 2:1.'
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'Find the equation of a line that is a tangent to the circle x^2+y^2=8 and perpendicular to the line 7x+y=0.'
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'When the line with slope -1 intersects region D, and the circle with center (3,2) has a distance to the line less than or equal to the radius 1. Find the maximum value of n in this case.'
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'Passing through point A(3,1), let P and Q be the points of contact of the two tangent lines that touch the circle x^{2}+y^{2}=5. Find the equation of the line PQ.'
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'For the circles \\( (x-5)^{2}+y^{2}=1 \\) and \ x^{2}+y^{2}=4 \, find: (1) How many common tangents do the two circles have? (2) Determine the equations of all common tangents for the two circles.'
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'Find the range of constant k such that the circle x^2 + y^2 - 4x - 6y + 9 = 0 (1) and the line y = kx + 2 have a point in common.'
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'Do the given circle and line have a point in common? If so, find the coordinates of that point.'
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'Given the radius and the radius representing an angle, let P be the intersection point with a circle of radius r.'
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'Reduce the given angle of 42140° to an angle on the unit circle in standard position.'
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'The centroid of the triangle formed by the moving point of distance 2 from the origin with fixed points (5,0) and (0,3) lies on the curve x^{2}+y^{2}-Ax-By+C=0.'
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'Find the following coordinates: (\x0crac{3}{14}, 0) and (0,-\x0crac{3}{4}).'
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'For triangle ABC with vertices A(1,1), B(2,4), and C(a, 0) to be a right-angled triangle, find the value of a.'
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'Shapes and Equations\nFor the three points A(0,0), B(2,5), C(6,0), find the coordinates of point P when PA² + PB² + PC² is minimized.'
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'Find the equation of the tangent line to the circle at point A (4,6) on the circle x^2 + y^2 - 2x - 4y - 20 = 0.'
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'In the coordinate plane, find the equations of the two lines that pass through the point (-2, -2) and are tangent to the parabola y=1/4 x^{2}.'
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'When the 3 lines 2x-y-1=0, 3x+2y-2=0, y=1/2x+k intersect at point A, what is the value of k, and what are the coordinates of point A?'
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'Find the situation of the three points A(-2, -2), B(2, 6), C(5, -3) on the coordinate plane.\n(1) Find the equation of the perpendicular bisector of segment AB.\n(2) Find the coordinates of the circumcenter of triangle ABC.'
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'Find the equation of the line passing through point (7,1) and tangent to the circle x^2+y^2=25, and find the coordinates of the point of tangency.'
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'For the circles given by (x-5)^2 + y^2 = 1 and x^2 + y^2 = 4, (1) how many common tangents do the two circles have in total? (2) Find all the equations of the common tangents for the two circles.'
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'A circle with its center on the line y = -4x + 5 and tangent to both the x-axis and y-axis, find the equation of the circle.'
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'Find the equation of the circle passing through the points (4, -1), (6, 3), and (-3, 0).'
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'Investigate the shape of the triangle ABC with vertices A(2a, a+√3a), B(3a, a), C(4a, a+√3a). Here, a>0.'
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'Find the coordinates of the point that divides the line segment AB between points A(x1, y1) and B(x2, y2) internally in the ratio m:n.'
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'Find the equation of the tangent line at the given point on the given circle.'
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'Find the coordinates of the remaining vertex D of the parallelogram with vertices A(4,5), B(6,7), C(7,3).'
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'Determine the values of a and b so that triangle ABC with vertices A(1,0), B(0,3), and C(a, b) becomes an equilateral triangle.'
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'On the number line, there are 3 points A(3), B(-3), C(5). Point D divides the line segment AB internally in the ratio 2:1, point E divides the line segment AC externally in the ratio 3:1, find the coordinates of the point that divides the line segment DE internally in the ratio 3:4.'
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'Let n≥2. There are n circles on a plane, where no two circles intersect each other and three or more circles do not intersect at the same point. How many intersection points are formed by these circles?'
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'Find the length of the line segment AB where A and B are the points of intersection of the parabola y=x^{2} (1) and the line y=x+3(2).'
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'Conversely, the point P(x, y) on the line x+y=2 satisfies AP^2-BP^2=4. Therefore, the required trajectory is the line x+y=2.'
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'Find the arc length and area.\n(1) Radius is 10 with angle π/5\n(2) Radius is 3 with angle 15°'
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'Find the coordinates of the remaining vertex D of a parallelogram with vertices A(5,-1), B(3,3), and C(-1,-3).'
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'When the point P (x, y) on the coordinate plane moves within the range of 3y ≤ x + 11, x + y - 5 ≥ 0, and y ≥ 3x - 7, find the maximum and minimum values of x² + y² - 4y.'
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'Find the locus of points P that satisfy the following conditions:\n(1) Points P equidistant from points O(0,0) and A(3,2)\n(2) Points P such that angle OPA = 90 degrees, where O(0,0) and A(6,0)\n(3) Points P such that AP^2 - BP^2 = 4, where A(3,2) and B(1,0)'
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'There are n circles on a plane, where any two circles intersect each other, and no three or more circles intersect at the same point. In how many parts do these circles divide the plane?'
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'Find the distance between the following two points: (1) A(-3), B(2) (2) A(-2), B(-5)'
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'Therefore, the coordinates of point C are , and thus . Also, the distance between point A and line (3) is . Therefore, the area of the triangle we seek is .'
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'In the xy-plane, considering the lines l: x+t(y-3)=0 and m: tx-(y+3)=0 as t varies over all real numbers, what kind of shape do the intersection points of lines l and m form? [Gifu]'
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'Regarding the triangle ABC with vertices A(1,1), B(2,4), and C(a,0)'
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'Graph the following equations representing lines on the coordinate plane.'
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'Find the equation of the tangent that satisfies the following conditions.'
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'Given three vertices A(4,5), B(6,7), C(7,3) of a parallelogram, find the coordinates of the remaining vertex D.'
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'Determine the shape of the triangle ABC with vertices A(2a, a+√3a), B(3a, a), and C(4a, a+√3a). Assume that a>0.'
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'What kind of triangle is triangle ABC with vertices A(1, -1), B(4, 1), and C(-1, 2)?'
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'Find the coordinates of the point which is symmetric to point with respect to the line .'
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'Find the equation of circle PR. (1) Passing through points (0,2), (-1,1) and with center on the line y=2x-8.'
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'Find the conditions for the general form of a circle equation x^{2}+y^{2}+l x+m y+n=0 to represent a circle. Also find the center and radius.'
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'Find the coordinates of the following points. (1) Point dividing the line segment 3:1 internally (2) Point dividing the line segment 3:1 externally (3) Point dividing the line segment 1:3 externally (4) Midpoint'
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'When the points A(a,-2), B(3,2), C(-1,4) are collinear, find the value of the constant a.'
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'Find the tangent at a point on the circumference of a circle.'
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'For points A(7,6), B(-3,1), and C(8,1) in the PR 3, let P be the midpoint of BC, Q be the point that divides CA externally in the ratio 3:2, and R be the point that divides AB internally in the ratio 3:2. Find the coordinates of the centroid of triangle PQR.'
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'Given the parabola y = x^2 and points A(1,2), B(-1,-2), C(4,-1), find the locus of points Q and R as point P moves on the parabola y = x^2. (1) Point Q that divides the segment AP in the ratio 2:1 (2) The centroid R of triangle PBC.'
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'Find the maximum value of k, such that the condition x^{2}+y^{2} ≤ 1 implies 3 x+y ≥ k.'
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'Find the locus of points P that satisfy the following conditions:'
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'Find the scale of the midpoint of the line segment connecting point 4 on the scale and point 9 on the scale.'
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'(1) In the coordinate plane, find all the equations of lines that are parallel to the line y=-2x and at a distance of √5 from the origin. [Tokyo Denki University] (2) Find the distance between the parallel lines 2x-3y=1 and 2x-3y=-6.'
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'Plot the regions represented by the following inequalities.'
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'Find the conditions of a and b such that the line y = ax + b has a point in common with the line segment connecting two points A(-1,5) and B(2,-1), and plot it on the ab plane.'
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'Find all values of a that divide the plane into 6 parts.'
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'Find the coordinates of the two intersection points of the two circles x^2 + y^2 = 10 and x^2 + y^2 - 2x + 6y + 2 = 0.'
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'1. Find the equation of a circle with center (0,0) and radius √2.'
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'Find the equation of the circle that has a center at (1,1) and is tangent to the line 4x+3y-12=0.'
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'On the coordinate plane, let A be the point (-3,2) and B be the point (4,0). Find the coordinates of the points that are equidistant from the x-axis and y-axis, respectively.'
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'Find the coordinates of the point that divides point A (x1, y1) and point B (x2, y2) externally. The external division ratio is m:n.'
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'Find the equation of a circle that passes through the point (2,3), is tangent to the y-axis, and has its center on the line y=x+2.'
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'In a triangle ABC, let D, E, F be the points dividing the sides BC, CA, AB in the ratio 3:1, respectively. Show that the centroid of triangle DEF coincides with the centroid of triangle ABC.'
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'Find the area of the triangle formed by the lines x-y+1=0, 2x+y-2=0, and x+2y=0.'
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'Determine the number of intersection points between a line and a circle in geometry and equations.'
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'Description question about the polar lines of a circle with respect to point A. Given a point A(p, q) outside the circle x^2+y^2=r^2, find the equation of the line β passing through the points of tangency P, Q of the two tangents drawn from point A to the circle. Using the equation p x+q y=r^2, prove that the polar line of point A passing through another point B implies that the polar line of point B passes through point A.'
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'Let S be the area of a regular pentagon inscribed in a circle of radius 1'
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'In the regular tetrahedron OABC with side length a, points P, Q, R are taken on edges AB, BC, and OC respectively. Starting from vertex O, passing through points P, Q, R in sequence, what is the length of the shortest path to vertex A?'
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'In quadrilateral ABCD, AB=4, BC=5, CD=t, DA=3-t (0<t<3). Also, assume that quadrilateral ABCD has a circumscribed circle.'
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'There are two lines that intersect the line y=x-1 and form an angle of 15 degrees, passing through the point (0,1). Find the equations of these lines.'
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'For a regular tetrahedron ABCD with edge length of 6, let E be the point on edge BC such that 2BE=EC, and let M be the midpoint of edge CD.'
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'Consider a triangle with side lengths a, a+2, a+4.'
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'In triangle ABC, AB = 8, AC = 5, and ∠A = 120°. Let D be the intersection of the angle bisector of ∠A and side BC. Find the length of segment AD.'
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'A cone with a radius of 2 and a slant height of 6 is tangent to a sphere O both on its lateral surface and at its center of the base. Determine the radius, volume, and surface area of this sphere.'
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'Choose the most appropriate term from (A) to (E) to fill in the blank.'
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'Practice 1: On the sides AB, BC, CA of an equilateral triangle ABC with side length 1, points D, E, F are taken such that AD=x, BE=2x, CF=3x. (1) Express the area of triangle DEF, S, in terms of x. (2) Find the value of x that minimizes S in (1) and the minimum value.'
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'In △ABC, where a=2, b=√2, c=1. Find:\n(1) cos B, sin B\n(2) The area of △ABC, S\n(3) The radius of the incircle of △ABC, r\n(4) The radius of the circumcircle of △ABC, R\nRefer to p. 265, Basic Concept 3, Basic 162.'
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'In the quadrilateral ABCD inscribed in a circle, with AB=4, BC=5, CD=7, and DA=10. Find: (1) the value of cos A (2) the area of quadrilateral ABCD.'
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'188\nMathematics I\n(1) As shown in the figure, take the vertices A, B, C of triangle T to be AB=5, BC=6, CA=7.'
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'Given b=4, c=4√3, and B=30°, find a, A, C, and R.'
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'In a right triangle where the sum of the lengths of the two sides is 20, find the right triangle with the minimum length of the hypotenuse, and determine the length of the hypotenuse.'
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'In triangle ABC, let the radius of the circumscribed circle be R. Find the following values: (1) When A=60°, C=45°, a=3, find c and R'
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'Practice (1) In the diagram on the right, find the lengths of line segments DE and AE.\n(2) Using the diagram on the right, find the following values: sin 15°, cos 15°, tan 15°'
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'Consider an equilateral triangle ABC with side length of 1. Points D, E, F are taken on sides AB, BC, CA respectively such that AD=x, BE=2x, CF=3x. (1) Express the area of triangle DEF, denoted by S, in terms of x. (2) Find the value of x that minimizes S and the minimum value.'
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'Please explain what a chart is based on the following text: In the Concise Oxford Dictionary (C.O.D.), a chart is described as a sea map of CHARTNavigator, including coast outlines, rocks, shoals, etc.'
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'In triangle ABC, let D be the intersection point of the angle bisector of angle A and edge BC. Find the lengths of segments BD and AD in the following cases:'
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'Find the area S of a regular octagon inscribed in a circle of radius a.'
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'Chapter 4 Geometry and Measurement\n15 Basics of Trigonometry\nProblem\nIf one angle of a triangle is 90 degrees and another angle is 30 degrees, find the measure of the remaining angle and trigonometric ratios.'
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'In △ABC, let the radius of the circumcircle be R. Find the following: (2) When a=√2, B=50°, and R=1, find the values of A and C'
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'How to represent the coordinates of point P(a, b) on the coordinate plane?'
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"To check if it is possible to draw more than four sets using circles, let's try to represent four sets A, B, C, D with circles. First, draw the Venn diagrams for sets A, B, C, and then try to add the Venn diagram for set D to observe the result.\n\nNext, to verify if we can draw four sets using circles, draw four different circles on a plane and count the intersections. In this case, the four circles must follow the following rules:\n- Any two circles intersect at two points\n- Any three circles do not intersect at the same point\n\nCalculate the number of regions created by the intersections of the four circles and check if this number matches the number of common parts formed by the four sets and their complements."
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'On the coordinate plane, there is one line and two parabolas, Line L: y=ax+b, Curve C_{1}: y=-2x^{2}, Curve C_{2}: y=x^{2}-12x+33. When Line L intersects Curve C_{1} and Curve C_{2} at two points each, the inequality a^{2}-a<b<a^{2} holds, where a>0.'
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'Find the area of the following figure. (2) \ \\mathrm{AB}=3, \\mathrm{AC}=3 \\sqrt{3}, \\angle \\mathrm{B}=60^{\\circ} \ parallelogram \ \\mathrm{ABCD} \'
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'Calculate the values of trigonometric functions from the given coordinates. (1) P(-1,1) (2) P(-√3, 1)'
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'In triangle ABC, when b=2, c=√5+1, and A=60 degrees, determine whether C is an acute angle, right angle, or obtuse angle.'
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"Define the conditions p, q, r regarding triangles as follows: p: all three internal angles are different q: not a right triangle r: no internal angle is 45 degrees Choose the correct options from each choice: (1) The contrapositive of the proposition 'r implies (p or q)' is 'a _______ implies not r'. [Choices] 0 (force and q) (1) (not p and not q) (2) (to be as q) (2) What triangle serves as a counterexample to the proposition '(p or q) implies r'? [Choices] Right isosceles triangle (1) A triangle with internal angles of 30 degrees, 45 degrees, 105 degrees (2) Equilateral triangle (3) Triangle with side lengths 3, 4, 5 (4) Isosceles triangle with a vertex angle of 45 degrees (3) r is the causal relationship for (p or q) as a _______. [Choices] (0) Necessary and sufficient condition (1) Only necessary but not sufficient condition (2) Only sufficient but not necessary condition (3) Neither necessary nor sufficient condition"
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'In triangle ABC, when the radius of the circumscribed circle is R, find the following: (1) Side c when C=120° and R=4. (2) Side b and radius R when A=45°, B=60°, and a=2.'
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'In an acute triangle ABC, let BD and CE be the perpendiculars dropped from vertices B and C to their respective opposite sides. If BC=a and the measure of angle A is represented by A, express the length of segment DE in terms of a and A. Also, you may use the property that if angle PRQ=90 degrees for segment PQ, then point R lies on the circumference of the circle with segment PQ as its diameter.'
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'There is a quadrilateral ABCD inscribed in a circle. If AB=4, BC=5, CD=7, DA=10, find the area S of quadrilateral ABCD.'
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'In the quadrilateral ABCD inscribed in a circle, with AB = BC = 1, BD = √7, DA = 2, find: (1) A (2) The length of side CD (3) The area of quadrilateral ABCD'
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'Determine the range of values for x such that the triangle with side lengths 3, 5, and x forms an acute triangle.'
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'Folded a piece of origami in the shape of an equilateral triangle with side length 10 cm, denoted as ABC.'
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'In a quadrilateral ABCD inscribed in a circle, with AB=8, BC=10, and CD=DA=3. Find the area S of the quadrilateral ABCD.'
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'In triangle ABC, if B=30°, b=√2, and c=2, find the values of A, C, and a.'
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'In triangle ABC where AB=6, BC=4, and CA=5, find the length of segment BD where the angle bisector of angle B intersects side AC at point D.'
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'In triangle ABC with AB=13, BC=15, and CA=8, let AD be the perpendicular from point A to side BC. Find the following values when AD is 90 degrees: (1) Length of BD, (2) sin angle B, (3) tan angle C'
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'On the line x=1, let the point with y-coordinate sqrt{3} be T. The point P where the line OT intersects the semicircle with radius 1 is the point in the figure. The angle θ we seek is ∠AOP.'
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'In triangle ABC, when a=5, b=7, and c=8, find angle B.'
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'In triangle ABC, where b=3, c=√2, and A=45°, find the length of side a.'
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'In the diagram on the right, find the lengths of the line segments AB, BC, and CA.'
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'Please explain the theorems and formulas related to the following triangle.'
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'Find the equation of the parabola after symmetrically moving the parabola y = -2x^2 + 3x - 5 with respect to the following line or point. (1) x-axis (2) y-axis (3) origin'
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'(1)\\[ \egin{aligned} A & =180^{\\circ}-(B+C) \\\\ & =180^{\\circ}-(30^{\\circ}+105^{\\circ}) \\\\ & =45^{\\circ} \\end{aligned} \\] Therefore, the area of triangle ABC is \\[ \egin{aligned} \\frac{1}{2} b c \\sin 45^{\\circ} & =\\frac{1}{2}(\\sqrt{6}-\\sqrt{2}) \\cdot 2 \\cdot \\frac{1}{\\sqrt{2}} \\\\ & =\\frac{\\sqrt{6}-\\sqrt{2}}{\\sqrt{2}} \\\\ & =\\frac{\\sqrt{2}(\\sqrt{3}-1)}{\\sqrt{2}} \\\\ & =\\sqrt{3}-1 \\end{aligned} \\]'
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'In a quadrilateral ABCD inscribed in a circle, with AB=2, BC=1, CD=3, and cos ∠BCD=-1/6. Find AD and the area of the quadrilateral ABCD.'
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'In a rectangle with a length of 40 cm, what is the minimum length of the diagonal? Also, what kind of rectangle is it at that time? If the vertical length of the rectangle is x cm, then the horizontal length is (20-x) cm. Additionally, x>0 and 20-x>0, hence 0<x<20. Let the length of the diagonal of the rectangle be l cm. l^2 = x^2+(20-x)^2 = 2x^2-40x+400 = 2(x-10)^2+200. The minimum value of l^2 occurs at x=10 with a value of 200. Since l>0, when l^2 is at its minimum, l is also at its minimum. Therefore, the minimum length of the diagonal is sqrt(200)=10 sqrt(2) cm. At this point, the horizontal length is 20-x=10 cm, making the diagonal length minimum when the rectangle is a square.'
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'Explain about the quadrants in a coordinate plane and give an example of the second quadrant.'
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'Radius of circumcircle and incircle of a triangle'
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'203 Basic triangle obtuse angle triangle conditions'
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'In quadrilateral ABCD, if AB=8, BC=5, CD=DA=3, and A=60 degrees, what is the length of the diagonal BD?'
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'The diagram on the right is a box plot of the scores of 30 students in a science test. When the scores that formed this box plot are represented in a histogram, which of the following 0 to 2 corresponds to it?'
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'In a rectangle with a perimeter of 40 cm, find the minimum length of the diagonal. Also, determine the characteristics of the rectangle at that time.'
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'A regular tetrahedron OABC with edge length 6 is given. Let L be the midpoint of edge OA, M be the point that divides edge OB into 2:1, and N be the point that divides edge OC into 1:2. Find the area of triangle LMN.'
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'There is a quadrilateral ABCD inscribed in a circle. If AB = 4, BC = 5, CD = 7, DA = 10, find the area S of quadrilateral ABCD.'
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'When the lengths of the three sides of ∆ABC are as follows, determine whether angle A is acute, right, or obtuse.'
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'On the semicircle with radius 1, the point where the x-coordinate is 1/2 is point P. The angle we seek is ∠AOP.'
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'In triangle ABC, where AB=3, AC=2, and ∠BAC=60°, if the angle bisector of angle A intersects BC at D, find the length of segment AD.'
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'Given the lengths of the diagonals AC and BD of the quadrilateral ABCD as p and q, and one of the angles they form as θ, express the area S of the quadrilateral ABCD in terms of p, q, and θ.'
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'In the quadrilateral ABCD, which is not a parallelogram, AD=BC. Let the midpoints of AB and CD be P and Q respectively, and the midpoints of AC and BD be M and N respectively. (1) Express →PQ, →MN in terms of →AD, →BC. (2) Prove that PQ is perpendicular to MN.'
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'A circle passing through point A(-3,0) and tangent to the line x=3 has center at P(x, y). Find the locus of point P.'
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'Consider the ellipse and the line and their intersection point . Express the ellipse excluding the point using the parameter .'
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"Ellipse\nIn this section, we will learn about the locus of points where the sum of distances from two fixed points remains constant.\nEquation of an ellipse\nIn a plane, the locus of points where the sum of distances from two fixed points F and F' remains constant. The two fixed points, F(c, 0) and F'(-c, 0) [c>0], are called the foci of the ellipse. Let's find the equation of an ellipse C with a sum of distances 2a from these two points using the concept of locus."
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'Let k be a constant. Determine the number of intersection points between the ellipse x^{2} + 4y^{2} = 20 and the line y = (1/2)x + k.'
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'Find the locus of point P, such that the ratio of distances from point TR (F(0,1)) and line l: y=-1 is given. 107 (1) 1: 1 (2) 1: 2 (3) 2: 1 Let P(x, y), and let PH be the perpendicular from P to line l, then PH=|y-(-1)|=|y+1|'
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'At the incenter of a triangle\n(1) In triangle ABC, where AB = 6, BC = 3, CA = 4, and the incenter is I. Express AI in terms of AB and AC.'
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"Let's consider the following curves and their trajectories. (1) Let C1 be a circle with radius 7 centered at the origin, and let C2 be a circle with radius 1 centered at point F(4,0). Let P be the center of the circle that is tangent to C1 and tangent externally to C2, then, what is the relation of P? Answer choices: (0) OP*FP is constant (1) |OP-FP| is constant (2) OP+FP is constant (3) OP^2+FP^2 is constant. Therefore, point P lies on an ellipse with foci O and F, and the length of the major axis determined."
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'If a circle with radius 2, x^2+y^2=25, is scaled by 3/5 in the x-axis direction with respect to the y-axis, what kind of curve will it become?'
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'Circle with center at (-2i) and radius 2 (1) Circle with center at (-1/2i) and radius 3/2'
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'What kind of shape does the parametric representation represent?'
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'Find the minimum distance between a point P on the ellipse x^2+4y^2=4 and a point Q on the line x+2y=3.'
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'Find the part obtained by removing point 2 from a circle with center at point 1 and radius 1.'
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'Chapter 4 Form and Curve - Simplifying to y^{2} = -12x, therefore, point P lies on the parabola y^{2} = -12x. Conversely, all points P(x, y) on this parabola satisfy the condition. Thus, the required trajectory is the parabola y^{2} = -12x.'
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'(1) Perpendicular bisector of the line segment connecting points 0 and 1 (2) Circle with radius 2 centered at point 3'
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'Find the minimum distance between the point P on the ellipse and the point Q on the line .'
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'Let TR be a constant. Find the number of intersection points between the ellipse x^2 + 4y^2 = 20 and the line y = \\frac{1}{2}x + k.'
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'Prove that when a line passing through the focus \ \\mathrm{F} \ of the parabola \\( y^{2}=4 p x(p \\neq 0) \\) intersects the parabola at points \\\mathrm{A}, \\mathrm{B}\, the product of the \ y \ coordinates of points \\\mathrm{A}\ and \\\mathrm{B}\ remains constant.'
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'Find the locus of point P such that the ratio of distances from point F(1,0) and line ℓ: x=-2 to point P is 1:2.'
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'Find the focus and directrix of the parabola . Also, sketch its general shape.'
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'For triangle ABC with vertices A(-1+i), B(2√3-1), and C(6+(√3+1)i), find the measure of angle BAC.'
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'Find the locus of the center P(x, y) of a circle passing through the point A(4,0) and tangent to the line x = -4.'
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'Given three different points A(α), B(β), C(γ) and the following relation between them, find the measures of the 86 angles in triangle ABC with these three points as vertices.'
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'TRAINING 41\nFrom point P(1,3), draw a perpendicular line to the line ℓ: 2x-3y+4=0, with the intersection point being H.\n(1) Find the coordinates of point H using vectors.\n(2) Find the distance between point P and line ℓ.'
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'Find the equation of a parabola with a focus and directrix as follows, and sketch its graph. (A) Point (-1,0), line x=1 (B) Point (0,2), line y=-2'
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'In the regular pentagon ABCDE with side length of 1, let AB = b and AE = e.'
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'【Example 30(2)】\n(Because it is not in the form of , eliminate from the condition expression.)\n, , , considering in the Cartesian coordinate plane\n\nLet so ...\nAlso, \nFrom (1), (2), \nThe region represented by (3) is the red part in the right of [Figure 5], so the range of the point is around and inside the parallelogram OCDB in [Figure 6].\nSolve example 16 on page 48 using coordinates\n, , , considering in the Cartesian coordinate plane, , , so , \n\nThe equation of line AD is \nThe equation of line BC is \nSolve (1), (2) to obtain , so \nTherefore, \n\nSimilarly, find the equation of lines , → find the coordinates of point and examine , can also be expressed in .\n\nIn this way, it is very interesting that the ratio of line segments can be obtained by simple calculations at the junior high school level. Furthermore, to avoid fractions appearing in the above calculations, it is permissible to continue with , .'
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'Represent points A, B in Cartesian coordinates as A(2 cos π / 6, 2 sin π / 6), B(4 cos π / 3, 4 sin π / 3)\nThat is, A(√3, 1), B(2, 2√3)\nTherefore, the Cartesian equation of the line AB is (2√3 - 1)(x - √3) - (2 - √3)(y - 1) = 0'
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'Practice 109 (1) The asymptotes are the two straight lines y=1/2x, y=-1/2x intersecting at the origin, hence the equation of the hyperbola to be determined is, where a>0, b>0,'
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'Find the standard form of the parabola with the focus at F(p, 0) (p ≠ 0) and the directrix line ℓ: x = -p.'
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'Find the polar equations of the following circle and line with respect to polar coordinates.135\n1) Circle with center at (1, 3/4π) and radius 1\n2) Line passing through point A(2, π/4) and perpendicular to line OA (O is the pole)\n3) Line passing through points A(2, π/6) and B(4, π/3)'
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'In the coordinate plane, with the origin O as the pole, and the positive part of the x-axis as the initial line. At this time, explain the relationship between the polar coordinates (r, θ) and the Cartesian coordinates (x, y) of the same point P.'
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'Find the polar equation of a circle with polar coordinates (a, 0) and radius a.'
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'Therefore, the equation of the trajectory of point P is x^{2}+y^{2}=a^{2}-1, representing a circle with radius \\sqrt{a^{2}-1} centered at the origin. However, it is necessary to exclude the 4 intersection points with the asymptotic lines y=\\pm a x, which are (\\pm \\sqrt{\\frac{a^{2}-1}{a^{2}+1}}, \\pm a \\sqrt{\\frac{a^{2}-1}{a^{2}+1}}) (with arbitrary signs).'
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'Important Example 139 The Use of Polar Coordinates\nProve that when the two ends of a chord passing through one focus \ \\mathrm{F} \ of an ellipse are \ \\mathrm{P}, \\mathrm{Q} \, \ \\frac{1}{\\mathrm{FP}} + \\frac{1}{\\mathrm{FQ}} \ is constant regardless of the direction of the chord.'
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'The polar equation with one focus F of a certain conic is represented as follows, with a as a positive constant and e as eccentricity.'
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'Translate the given text into multiple languages.'
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'Practice with z being a non-zero complex number. When the point z-1/z moves along the line segment connecting the points i and 10/3 i, illustrate the range of 88 points with real parts in the complex plane.'
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'Let P(z) be the incenter of triangle OAB with vertices at three distinct points O(0), A(α), and B(β), then show that z satisfies the following equation.'
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'Consider the given text in the context of multiple languages.'
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'Please express the condition for points A, B, and C to be collinear mathematically.'
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"The locus of a point P, where the difference in distances from two distinct fixed points F and F' is a non-zero constant, is called a hyperbola, with the points F and F' as its foci. The difference in distances should be less than the length of the segment FF'. Let's find the equation of a hyperbola with foci at the points F(c, 0) and F'(-c, 0) and a distance difference of 2a from these two points. Here, c > a > 0."
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'Find the coordinates of the midpoint and the length of the chord formed by the intersection of the line y=x+2 and the ellipse x^{2}+3y^{2}=15.'
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'The plane we are looking for passes through the point C(1,3,-2) and is perpendicular to the (a) z-axis, (b) x-axis, and (c) y-axis.'
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'Example 137 Polar coordinates and trajectory Let the polar coordinates of point A be (2,0). Let Q be any point on the circumference of the circle C whose diameter is the line segment OA connecting the pole O and point A. At point Q, draw the tangent line of circle C and drop a perpendicular OP from the pole O to point P. Let the polar coordinates of point P be (r, θ). Find the polar equation of the trajectory of point P, where 0 ≤ θ < π.'
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'(1) Find the equations of the lines passing through point A(-2,3) that are parallel and perpendicular to the line ℓ: 5x+4y-20=0.'
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'(a) Focus: point (-1/2, 0), Directrix: line x = 1/2, diagram omitted'
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"Let F be the point (3,0) and the center of the circle be P. The circle with radius 2 and center (-3,0) is denoted as F'. Since the radius of circle C is the segment PF, when the two circles are tangent, PF' = PF + 2. Therefore, PF' - PF = 2, which means the point P is on a hyperbola with foci at F'(-3,0) and F(3,0) where the distance between the foci is 2. The equation of this hyperbola is given by \\( \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0) \\). From the coordinates of the foci, we have \ a^{2}+b^{2}=3^{2} \ and from the difference of distances from the foci, we get \ 2a=2 \, which gives us \ a=1 \. Thus, \ b^{2}=9-a^{2}=8 \. Therefore, point P moves along the hyperbola x^{2}-\\frac{y^{2}}{8}=1. However, since PF' > PF, we have x > 0, and thus the trajectory we seek is the part of the hyperbola x^{2}-\\frac{y^{2}}{8}=1 where x > 0."
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'Example 1 | Basic Vectors\nDetermine the following vectors using the vertices of the regular hexagon ABCDEF with side length 1 as shown in the figure and the intersection point O of the diagonals AD, BE:\n(1) Vector equal to AB\n(2) Vector with the same direction as OA\n(3) Inverse vector of AC\n(4) Vector parallel to AF and magnitude 2'
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'Practice For a triangle ABC and any point P in the plane, the following vector equation represents a circle. What kind of circle is it?\n(1) |→BP+→CP|=|→AB+→AC|\n(2) 2→PA⋅→PB=3→PA⋅→PC'
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'Consider a line segment AB of length 2, where point A lies on the x-axis and point B moves along the y-axis. Find the locus of point P such that on extending the line segment AB, BP = 1.'
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'Find the polar equation of a circle with center at the pole and radius a.'
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'Prove that when the tangent at point P(x1, y1) on the parabolic curve y^2=4px(p>0) intersects the x-axis at T and the focus of the parabola is at F, then ∠PTF=∠TPF. Given that x1>0, y1>0.'
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'In mathematics, given that OP = r, OA = 2, and angle AOP = |θ - π / 4|, from (1),\nr cos |θ - π / 4| = 2\nwhich means r cos (θ - π / 4) = 2'
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'Find the equation of the tangent line at point P(x, y) and determine the condition for the midpoint of the line segment drawn perpendicular to X=0 from that point to pass through the origin.'
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'Practice\n\nLet a > 0. One end of a string with length 2πa is fixed at the point A(a, 0) on the circle x^{2}+y^{2}=a^{2} with radius a, and it is wound clockwise around the circle. When stretching the string and removing it from the circle, find the length of the curve traced by the other end P of the string.'
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'Since AQ=BQ=CQ, it follows that AQ²=BQ²=CQ². Let Q(x, 0, z), then from AQ²=BQ² we have (x-2)²+1+(z+2)²=(x+2)²+(z-1)² which leads to 4x-3z=2. Also, from BQ²=CQ² we have (x+2)²+(z-1)²=(x-3)²+1+(z+3)² which leads to 5x-4z=7. By multiplying equation (3) by 4 and equation (4) by 3, we get x=-13. Substituting x into equation (3) gives z=-18. Therefore, Q(-13,0,-18).'
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'(1) Successive circles with any radius centered at the origin, x^{2}+y^{2}=5'
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'When the circle x^2+y^2=4 is scaled or enlarged in the following ways, what kind of curve does it become? (1) Scaled to 1/2 of its size in the y-direction relative to the x-axis (2) Enlarged to 3 times its size in the x-direction relative to the y-axis'
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'Find the shape of the curve when the tangent is perpendicular to OP, and derive the equation when the curve passes through the point (2,1).'
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'Let A(r1,θ1) and B(r2,θ2)[r1 > 0, r2 > 0]. Find the area of triangle OAB, denoted as ΔOAB.'
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'When the circle C with a radius of a/4 rotates without sliding, being tangent to the circle O with a radius of a centered at the origin O, and assuming that the fixed point P on the circle C was initially located at the fixed point A(a, 0) on the circumference of the circle O, express the curve (astroid) traced by P with a parameter θ. Here, θ is the angle formed by the line segment connecting the center C and O of circle C with the positive direction of the x-axis.'
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'Consider a circle C with endpoints of a diameter at A(3, -5) and B(-5, 1)'
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'Find the distance between the point A(2√3, 2√3, 6) and the plane x + y + z - 6 = 0, and using the method of the midpoint, determine the radius of the sphere.'
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'Since region D is the shadow part in the right figure, let P be the point of intersection of the circle x²+(y-1)²=1 and the line ℓ excluding the origin, and let Q be the point of intersection of the line x = √2/3 and the line ℓ. The value of L = PQ = OP - OQ needs to be calculated.'
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'What equation represents the trajectory of a point (P) with polar coordinates (a, 0) as the eccentricity e in the polar coordinate system?'
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'Translate the given text into multiple languages.'
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'When the plane (1) and the spherical surface S are tangent, find the coordinates of point P and calculate the maximum value that P can take from the given conditions.'
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'In the tetrahedron OABC, points P are on edge OA, points Q are on edge AB, points R are on edge BC, and points S are on edge CO. When the four points are connected in this order to form a figure that is a parallelogram PQRS, prove that the intersection point of the two diagonals of this parallelogram PQRS lies on the line segment connecting the midpoints of AC and OB respectively.'
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'Find the vector equation of a straight line passing through the centers of circles C1 and C2.'
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'Reissue Problem 127 Properties of Tangents of Hyperbolas The tangent at point on the hyperbola intersects the asymptote line at points . Let be the origin. Prove that the area of triangle is independent of the choice of point . [Similar to Tokyo University]'
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'Let the minimum value of the distance between a point on the curve and a fixed point be a positive number . In the plane, consider the point , the hyperbola given by , and the hyperbola given by . [Similar to Okayama University] (1) When point is on , find the point that minimizes and its minimum value. (2) When point is on , find the point that minimizes and its minimum value.'
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'Mathematics C 185 The common range with x > \\ frac {5}{2} is x > 3, so the required trajectory is the part of the hyperbola (x-2)^{2}-y^{2} / 3=1 where x > 3, y > 0. When plotted, it looks as shown in the figure on the right.'
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'Given a line segment AB of length l (>0), where endpoint A lies on the x-axis and endpoint B moves along the y-axis. Find the locus of the point P that divides the line segment AB internally in the ratio m: n. Here, m>0, n>0, and m≠n.'
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'(1) \\( (x+1)^{2}+(y+2)^{2}=25 \\)\n(2) Proof omitted, \ 3 x+4 y-14=0 \'
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'Let the circle x^2+y^2=1 be C0, and the ellipse x^2/a^2+y^2/b^2=1 (a>0, b>0) be C1. For any point P on C1, what is the necessary and sufficient condition in terms of a, b for the existence of a parallelogram with P as a vertex that is circumscribed about C0 and inscribed in C1?'
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'From point A(4,5), draw a perpendicular line to the line ℓ: x+2y-6=0 and mark the point of intersection with ℓ as H.'
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'Find the equation of the normal to the circle C2 at the center coordinates q, -q'
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'(1) Focus: point (0, -1/8), Directrix: line y = 1/8, diagram omitted'
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'In the xy-plane, let O(0,0) and A(1/√2, 1/√2). Find the area of the locus of points P that satisfy (PA ⋅ OA)^2 + |OP - (OP ⋅ OA)OA|^2 ≤ 1.'
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'Find the trajectory of point P that satisfies the following conditions:\n(1) The distance ratio between point F(4,2) and the line x=1 is 1:√2 for point P\n(2) The distance ratio between point F(0,-2) and the line y=3 is √6:1 for point P'
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'Illustrate the regions represented by the following inequalities:'
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'Point P lies on the line OG and on the plane ABD. Find k and determine the coordinates of point P.'
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'Since | u |=1, then x²+y²=1. Substituting (1) into (2), (3y)²+y²=1, hence 10y²=1. Therefore y=±1/√10. From (1), x=±3/√10 (same sign). Therefore find u=(3/√10, 1/√10), (-3/√10, -1/√10).'
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'(1) Right triangle with \ \\angle \\mathrm{A} = 90^{\\circ} \.\n(2) Equilateral triangle.'
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'Find the conditions for point Q to lie on the plane alpha, and derive the equation of the plane alpha.'
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'When each side of rectangle ABCD is tangent to ellipse E, let the angle between OA and AB be θ. Express the area of rectangle ABCD in terms of θ.'
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'Find the standard form equation of a parabola with focus at F(0, p) (p ≠ 0) and directrix as the line ℓ: y = -p.'
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'Prove the condition when triangle ABC is an isosceles triangle with AC=BC.'
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'Find the coordinates of the point dividing the line segment AB externally in the ratio m:n.'
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'Find the equations of the tangents at points P and Q on the curve.'
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'What is the condition for the two lines AB and AC to be perpendicular?'
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'Right-angled isosceles triangle with \ \\angle \\mathrm{A} = 90^{\\circ} \'
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"Find the equation of the ellipse with foci at (1,0) and F'(-1,0)."
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"Since points O, A', B' are on the xy plane, the figure formed by the intersection of the sphere S and the xy plane is a circle passing through O, A', B'. The equation representing this circle is (x - 5/6)^2 + (y - 5/6)^2 = 25/18, z = 0, therefore the coordinates of the center of the circle are (5/6, 5/6, 0)."
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'Prove that the midpoints of the sides AB, BC, CD, DA of the quadrilateral ABCD are P, Q, R, S respectively, and the midpoints of the diagonals AC, BD are T, U, then the midpoints of the segments PR, QS, TU are the same.'
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'Reference: When the solutions are plotted on the complex plane, the points z₀, z₁, z₂, z₃ are the vertices of a square inscribed in a circle of radius 2 with the origin O as the center.'
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'(4) The center is the point 1+sqrt(3)i, and the radius is sqrt(3)'
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'Derive the equation of the line passing through the points (a, 0) and (0, b) [a≠0, b≠0].'
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'On a number line, point A(a-1) and B(a+2) are connected to form line segment AB, which is internally divided into the ratio 2:1, with points C and D as the division points. (1) Find the distance between points C and D. (2) Determine the value of a for point E(-1) to be the midpoint of line segment CD.'
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'In triangle ABC with vertices A(a1, a2), B(b1, b2), C(c1, c2), let D, E, F be the points that divide sides BC, CA, AB in the ratio m:n respectively. Here, m>0, n>0. (1) Find the coordinates of points D, E, F. (2) Prove that the centroid of △DEF coincides with the centroid of △ABC.'
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'Find the distance between the following points and lines.'
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'When the three lines 4x + 3y - 24 = 0, x - 2y + 5 = 0, and ax + y + 2 = 0 intersect at one point, find the value of the constant a.'
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'Find that point. Find the coordinates of the remaining vertex D of the parallelogram with vertices A(1,2), B(5,4), C(3,6).'
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'Consider the two conditions p:(x-1)^{2}+(y-1)^{2} ≤ 4, q:|x|+|y| ≤ r. Here, r > 0. Find the range of values for the constant r that makes q a sufficient condition for p.'
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'Find the equation of the following circles: (1) Center at (3, -2), radius of 4 (2) Center at (0, 3) and passing through (-1, 6) (3) With endpoints of diameter at (-3, -4) and (5, 8)'
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"Please indicate the page number of Apollonius' circle from the index."
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'There are two points A(-1,3) and B(5,11) on the plane.'
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'Example (12) Division and internal division and external division on the number line'
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'Considering an equilateral triangle with a side length of 2 and one vertex on the x-axis, let the coordinates of the vertices be (a, 0), (b, 1), and (b, -1). As the centroid coincides with the origin, we have a=-2b. With a side length of 2, we can use the equation (b-a)^2 + (1-0)^2 = 2^2 to get (b-a)^2=3 (1), substituting (1) into (2) gives 9b^2=3. Due to the symmetry of the equilateral triangle, when b= ± √3/3, we have a= ∓ 2√3/3, b= ± √3/3, resulting in the vertices (2√3/3, 0), (-√3/3, 1), (-√3/3, -1) (in the same order) or (-2√3/3, 0), (√3/3, 1), (√3/3, -1)'
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'Practice (1) Find the equation of a circle with center on the line y=x, tangent to the line 3x+4y=24 and the coordinate axes. Find 101. (2) Find the equation of a line with slope -1 that is tangent to the circle x^{2}+2x+y^{2}-2y+1=0.'
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'In the presence of the parabola , divide the plane into two regions.\n③(1) Find the conditions for the points and to not lie on the same parabola but belong to different regions. Furthermore, illustrate the region represented by all points satisfying those conditions on the plane.\n(2) Determine the range of values that can take when satisfy the conditions obtained in (1). [Aichi University of Education]'
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'For a circle of radius r (x-p)^{2}+(y-q)^{2}=r^{2} and a line lx+my+n=0, the coordinates of the intersection points of the circle and the line can be determined as real solutions of the simultaneous equations (1), (2). Also, when the intersection point is a point of tangency, the solution becomes a repeated root.'
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'The maximum value is √3 when θ = 2/3 π, and the minimum value is -√3/2 when θ = 0.'
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'Find the value of the constant m when the area of the triangle OAB with vertices O(0,0), A(4,0), and B(2,2) is bisected by the line l: y = m(x + 1) + 1.'
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'In the xy-plane, there are two points A(3,2) and B(8,9). When point P moves along the line ℓ: y=x-3, find the minimum value of AP+PB and the coordinates of point P at that time.'
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'Centroid of a Triangle\nFind the centroid of triangle ABC with vertices A(x_{1}, y_{1}), B(x_{2}, y_{2}), and C(x_{3}, y_{3}).'
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'Let points A and B be A(-1, 5) and B(2, -1) respectively. For real numbers a and b, suppose the line y=(b-a)x-(3b+a) shares a point with the segment AB consisting of 3121 segments. Illustrate the region where point P(a, b) exists.'
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'On the xy plane, with the origin O and point A(2,0) given, let triangle OAB be an equilateral triangle with point B in the first quadrant. Furthermore, inside triangle OAB, a point P(a, b) is taken, and perpendiculars PL, PM, PN are dropped from P to sides OA, AB, BO respectively. (1) Find the coordinates of point B. (2) Find the value of PL+PM+PN.'
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'Let the vertex of the parabola y=x^{2}-x be P. Point Q is a point on this parabola, distinct from the origin O(0,0) and point P. If ∠OPQ is a right angle, find the coordinates of point Q.'
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'Find the equation of the line ℓ that touches both curves C1: y=(x-1/2)^2-1/2 and C2: y=(x-5/2)^2-5/2.'
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'Prove that the triangle ABC with vertices A(4,5), B(1,1), and C(5,-2) is a right isosceles triangle.'
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"Let A(1,4), B(-2,-1), C(4,0). Denote the symmetric points of B and C with respect to point P(a, b) as B' and C'. Prove that the centroid G' of triangle A'B'C' is the symmetric point of the centroid G of triangle ABC with respect to point P."
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'Passing through points A(3,0) and B(5,4) and with center at (2,3), what is the radius of circle C1? The circle C2 is symmetric to circle C1 with respect to the line AB. What are the coordinates of the center of circle C2? Also, if P and Q are points on circles C1 and C2 respectively, what is the maximum distance between point P and point Q?'
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'Practice There are two points A(-1,3) and B(5,11) on the plane.'
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'Find the coordinates of the remaining vertex D of the parallelogram with vertices A(3, -2), B(4, 1), and C(1, 5).'
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'Illustrate the locus of the point (x+y, x-y) as the real numbers x and y vary satisfying the following conditions: (2) x^2 + y^2 ≤ 4, x ≥ 0, y ≥ 0'
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'Find the locus of points at a constant distance r from a fixed point O.'
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'Find equations of circles passing through tangents'
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'Plot the region where the line y = -4tx + t^2 - 1 passes through as t varies from -1 to 1.'
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'Let s, t be real numbers such that s < t. Let A(1,2), B(s, s^2), C(t, t^2) be three points on the coordinate plane that are collinear. Determine the relationship between s and t.'
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'Find the value of the constant m when the area of the triangle OAB with vertices O(0,0), A(4,0), and B(2,2) is bisected by the line y=mx+m+1.'
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"Basic Concepts 12 relationships of circles (Mathematics A learning topics) The distance between the centers of two circles with radii r and r' (r>r') is d. Select the appropriate option from the following positional relationships: 1. Externally to each other 2. Tangent externally 3. Intersecting at 2 points 4. Incidentally tangent 5. One inside the other"
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'Find the equation of a line that satisfies the condition for a line to be tangent to a circle.'
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'The coordinates of the center are (0, \\frac{r^{2}+1}{2}), the coordinates of the tangent points are (-\\sqrt{r^{2}-1}, \\frac{r^{2}-1}{2}), (\\sqrt{r^{2}-1}, \\frac{r^{2}-1}{2}), where r > 1'
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'Does the circle x^2 + y^2 = 5 intersect with the following lines? If so, find the coordinates of the points.'
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'Find the values of the constant a for which the three lines (x-axis, y=x, (2a+1)x+(a-1)y+2-5a=0) do not form a triangle.'
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'Length and area of 4 sectors Radius r, central angle θ (radians) for a sector (1) Arc length l l=rθ (2) Area S S=1/2 r^2 θ=1/2 rl'
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'Prove that the two lines passing through points Q and R from the point (b, c) outside the circle pass through point P.'
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'Explain the definition of a general angle. For example, when a general angle θ is θ = 400°, indicate what angle it is actually equivalent to.'
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'The coordinates of the point P that divides the line segment connecting point A (-1, -3) and point B into the ratio 2:3 at (1, -1) are given. Find the coordinates of point B.'
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'Find the equation of the line passing through the point (2,-3) and intersecting the circle x^2+y^2=10 at two tangent points.'
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'Let 12 be constants a, b (a>b>0), and take two points A(0, a) and B(b, 0) on the xy plane. Point P is a point around and inside the square F with side AB. When the origin O(0,0) is outside the square F, express the following in terms of a, b: (1) Coordinates of the other 2 vertices of square F besides A, B (2) Maximum length of segment OP (3) Minimum length of segment OP'
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'In triangle ABC with vertices A(a1, a2), B(b1, b2), C(c1, c2), let D, E, F be the points that divide sides BC, CA, AB in the ratio m:n. Here, m is greater than 0 and n is greater than 0.'
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'(2) Let a be a constant, and a > 1. Let the point P(a, t) (t is a real number) on the line ℓ: x=a pass through the two tangents of the circle C: x^{2}+y^{2}=1, with contact points A and B respectively. Prove that the line AB does not depend on the point P, and find the coordinates of the fixed point.'
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'(2) Find the locus of points equidistant from points A and B.'
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'Find the equation of a circle that touches both the x-axis and the y-axis, and passes through the point (2,1).'
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'Explain about the circle passing through two intersection points of circles, and the line.'
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'Find the equation of the line passing through point P(2,1) and tangent to the circle x^{2}+y^{2}=1.'
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'The ratio of the distance from a point to the line x+y-1=0 and the distance to the line x-y-2=0 is 2:1. Find the equation of the trajectory formed by such points.'
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'176 110 Trajectory of the centroid of a triangle (interactive form) 2 points A (6,0), B (3,3), and a moving point Q on the circle x^2 + y^2 = 9 form a triangle with centroid P.'
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'Find the equation of the circumscribed circle of triangle ABC with vertices A(-2,6), B(1,-3), and C(5,-1).'
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'Find the equation of the tangent line at the given point for the following circles:'
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'Coordinates of points\nLet point A(x1, y1), point B(x2, y2), point C(x3, y3) be given.'
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"Let's consider how the values of the maximum and minimum of the -intercept change based on the value of the slope when moving the line in parallel to have a common point with the boundary line or tangent to a line or a circle."
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'Find the coordinates of the point P on the x-axis that is equidistant from points A(1, -2) and B(-3, 4).'
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'Given that point A is (2,1) and its symmetrical point is B, and the symmetrical point of B with respect to the line y = 2x - 3 is C, with coordinates (-1,3), find the coordinates of point A.'
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'Find the equation of the line joining the two points of contact of the two tangents drawn from point (2,-3) to the circle x^2+y^2=10.'
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'Let a>b>0. Let P be the point of intersection between the tangent at point (b, sqrt(a^2-b^2)) on the circle x^2+y^2=a^2 and the x-axis. Furthermore, draw two tangents from a point (b, c) outside the circle to the circle, and let the points of contact be Q, R. Show that the line passing through points Q and R also passes through point P.'
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'Let a, b (a>b>0) be constants, and take two points A(0, a) and B(b, 0) on the xy plane. Point P is a point on or within the square F with side AB. When the origin O(0,0) is outside the square F, express the following in terms of a, b.'
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'Practice Point P(1,2) and lines l: 3x+4y-15=0, m: x+2y-5=0. (1) Regarding line l, find the coordinates of the point Q which is symmetric to the point P about line l. (2) Regarding line l, find the equation of the line which is symmetric to line m about line l.'
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'(1) When three different points (1,1), (3,4), (a, a^2) are collinear, find the value of the constant a.'
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'When k varies over all real numbers, what geometric figure is formed by the intersection of the two lines l1: ky+x-1=0 and l2: y-kx-k=0?'
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'How to find the tangent line l at point P(x1, y1) on the circle C:(x-a)^{2}+(y-b)^{2}=r^{2}(r>0)'
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'Find the distance between two points on a coordinate plane'
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'Find the equation of the incircle of the triangle enclosed by the lines x=3, y=2, and 3x-4y+11=0.'
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'Consider the two circles and determined by the following equations.\C_{1}: x^{2}+y^{2}=4, \\quad C_{2}: x^{2}-6 r x+y^{2}-8 r y+16 r^{2}=0\ (1) Find the center coordinates and radius of . (2) Determine the value of when and are tangent. (3) Find the value of when the radii of the two circles are equal, and find the equation of the line passing through the two intersection points.'
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'The maximum value of 165 is 16, and the coordinates of point P are (5/√26, 1/√26) or (-5/√26, -1/√26)'
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'Consider the tangents at points P(0,3) and Q(6,15) on the parabola C: y=x^{2}-4 x+3 as l and m respectively. Find the area of the region enclosed by these two tangents and the parabola. Angle basics 246,247'
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'Find the equation of the common tangent for the circles and .'
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'On a number line, find the distance between the following points:\n(1) Origin O and point P(a)\n(2) The distance between two points A(a) and B(b) AB'
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'Circle C: x^{2}+y^{2}-4 x-2 y+4=0 is tangent to the circle D with center (-1,1).'
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'Find the equation of the circle passing through point A(8,6) and tangent to the y-axis with the smallest radius.'
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'Practice: Find the locus of points equidistant from points A(2,3) and B(6,1). Also, find the locus of points Q where the ratio of distances is 1:3.'
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'In the triangle ABC consisting of the point A(3,1) on the xy plane, the point B on the x-axis, and the point C on the line y=x, the set S is defined as all triangles ABC where the sum of the sides AB + BC + CA is equal to 0. The values for the x-coordinates of B and C that minimize the perimeter are when B has a value of X and C has a value of Y, and at this point the perimeter is AB + BC + CA = Z.'
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'Plot the moving region of the point (x+y, x-y) as real numbers x, y vary satisfying the following conditions: (1) -1 ≤ x ≤ 0, -1 ≤ y ≤ 1 (2) x^2 + y^2 ≤ 4, x ≥ 0, y ≥ 0'
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'Find the equation of the circumscribed circle of the triangle with vertices (-2, -1), (4, -3), and (1, 2).'
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'(1) Find the equation of the circle passing through the two intersections and the origin O: with center (-5, 5) and radius 5. (2) For any constant k, the circle x^2+y^2-2kx-4ky+16k-16=0 passes through two points: (k, 0) and (0, 4-k).'
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'Find the locus of the centroid P of a triangle with three vertices Q moving on the circle x^2+y^2=9, and with fixed points A(3,0) and B(0,3).'
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'A line passing through the point A(2,1) intersects the circle C: x²+y²=2 at two distinct points P and Q, and the length of the line segment PQ is 266. Find the equation of the line.'
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'Find the coordinates of the remaining vertex D of the parallelogram with vertices A(3,-2), B(4,1), and C(1,5).'
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'Let a be a constant greater than 1. Let P(a, t) (where t is a real number) be a point on the line ℓ: x = a, and let A and B be the points of tangency of two tangents to the circle C: x^{2} + y^{2} = 1. Show that the line AB passes through a fixed point regardless of the location of point P, and find the coordinates of that fixed point.'
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'Two points A and B move on the parabola y=x^{2} in the xy-plane. The line segment connecting them to the origin O forms triangle AOB, where angle AOB = 90 degrees. Find the locus of the centroid G of triangle AOB.'
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'In triangle ABC, given that ∠BAC=θ, AB=sinθ, and AC=|cosθ|. Also, ensure that θ≠π/2. Find the maximum and minimum values of BC^2.'
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'Find the equation of the line that passes through the point (1, √3) and forms an angle of π/3 with the line y=-x+1.'
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'For points A(5,4), B(0,-1), C(8,-2), let P be the point which divides the line segment AB externally in the ratio 2:3, and Q be the point which divides the line segment AB externally in the ratio 3:2. G is the centroid of triangle ABC.'
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'115 yen \ x^{2}+y^{2}=1 \ excluding the point \ -1,0 \'
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'Through the point A(8,6) and tangent to the y-axis, find the equation of the circle with the smallest radius.'
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'Find the equation of the common tangent to the circles C1: x ^ 2 + y ^ 2 = 9 and C2: x ^ 2 + (y - 2) ^ 2 = 4.'
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'Find the equation of a parabola with focus at point (2,0) and directrix at line x=-2. Also, sketch its graph.'
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"On the xy plane, there are circles C₁: x^2 + y^2 - 2x = 0, C₂: x^2 + y^2 - x = 0. For a moving point P on circle C₁ excluding the origin O, let Q be the intersection point of line OP and circle C₂ that is not O, and let Q' be the point symmetric to Q with respect to the x-axis. Find the equation representing the trajectory of the midpoint M of line segment PQ', and illustrate its general shape."
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'Please explain the difference between the ellipse () and the ellipse with its foci on the y-axis ().'
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'Find the locus of the center P of the circle that is tangent to both the circle (x-4)^{2}+y^{2}=1 and the line x=-3.'
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'(1) Find the coordinates of point P on the hyperbola that minimizes the distance between point P and point A(0,2), and determine the minimum distance at that point.'
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'Find the locus of the centers of the circles passing through the point (3,0) and touching the line x=-3.'
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'When the curve and the line have a common point in the first quadrant, and the tangents of the two curves at that point are the same, find the value of the constant .'
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'Since the line PQ is parallel to the x-axis, the area of △APQ is 1/2⋅{3/2-(-3/2)}⋅{1/2-(-1)}=9/4. Another approach is to consider that the area of △APQ is maximized when the distance d between point Q and line AP is maximized. Let Q(√3cosθ, sinθ)(0 ≤ θ < 2π) be (*), then the equation of line AP is x-y-1=0. Thus, d=|√3cosθ-sinθ-1|/√1^2+(-1)^2=1/√2|2sin(θ+2/3π)-1|. Therefore, sin(θ+2/3π)=-1, which means θ+2/3π=3/2π. When θ=5/6π, d reaches the maximum value of 3/√2. At this point, Q(-3/2, 1/2), △APQ=1/2⋅AP⋅3/√2=1/2⋅3/√2⋅3/√2=9/4.'
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'Point T divides line segment PQ in the ratio 1:2, so x=\\frac{2 \\cdot a+1 \\cdot 0}{1+2}, y=\\frac{2 \\cdot 0+1 \\cdot b}{1+2}. Therefore, a=\\frac{3}{2} x, b=3 y. Substituting these into (1) gives \\frac{9}{4} x^{2}+9 y^{2}=1, hence 9 x^{2}+36 y^{2}=4. Therefore, the locus of point T is the ellipse 9 x^{2}+36 y^{2}=4 and its general shape is shown on the right.'
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'Find the equation of the tangent line drawn from point (-1,3) to the ellipse using the discriminant of the quadratic equation.'
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'Find all the values of z for which the two straight lines OA and OB intersect perpendicularly.'
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'In the right isosceles triangle ABC, where AB = AC and BC = 2, find the maximum area of an ellipse that touches each side. You can use that the area of an ellipse with major and minor axes of lengths 2a and 2b respectively is πab. (Similar to a question from the University of Tokyo)'
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'Find the locus of the center of the circle passing through the point (3, 0) and tangent to the line x = -3.'
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'In the case of parallelogram OACB, the midpoint M of side AB lies on the real axis. Furthermore, since point C lies on the line OM, point C is also on the real axis. Therefore, prove that w is a real number or a pure imaginary number.'
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'Find the minimum distance between a point P on the parabola y²=6x and a fixed point A(a,0), where a is a real constant.'
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'Find the focus and directrix of the parabola x^2 = -8y, and sketch its rough shape.'
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'On the ellipse, let there be a point P that lies on the major axis but not on the minor axis, and two lines connecting the ends of the minor axis intersect the major axis or its extension at points Q and R. If O is the center of the ellipse, prove that the product of the lengths of line segments OQ and OR is constant.'
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'Find the locus of points P that satisfy the following conditions:\n(1) The distance ratio between point F(1,0) and the line x=3 is 1:√3\n(2) The distance ratio between point F(3,1) and the line x=4/3 is 3:2'
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'Show the conditions for points A, B, and C to have AB and AC perpendicular.'
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'Let a>2, b. Circle C with radius b rotates without slipping while internally tangent to the fixed circle O with center at origin O and radius a. The fixed point P(x, y) on circle C was initially at the fixed point A(a,0) on the perimeter of circle O. Define the rotation angle θ from the positive x-axis of the line segment connecting center C of circle C and origin O. Express the curve traced by point P in terms of the parameter θ. Assume point P(x, y) on circle C tangent to circle O was initially at point A(a,0), and it is at the position shown in the diagram when ∠COA = θ.'
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'Please convert the following points to polar coordinates: (1, √3), (-2, -2), (-3, √3)'
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'Given that the Cartesian coordinates of point P are (x, y) and the polar coordinates are (r, θ), prove the following relationships.'
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'Find the equations of the tangent lines at points P and Q on the following curves:'
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'A regular triangle T1 is inscribed in a circle S1 with radius 1. Let S2 be the circle inscribed in T1, and U1 the square inscribed in S2. Furthermore, let S3 be the circle inscribed in U1, T2 be the regular triangle inscribed in S3, and so on, inscribing circles S4 and squares U2 accordingly. In this manner, a sequence of circles S1, S2, S3, ..., a sequence of regular triangles T1, T2, T3, and a sequence of squares U1, U2, U3 is constructed.'
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'Find the trajectory of the point P(x, y) where the difference in distance from points A1 and A2 is 6, and plot this trajectory on the xy-coordinate plane.'
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'When point P (x, y) moves one round counterclockwise along the circle with radius 1 centered at the origin, points Q1 (-y, x) and Q2 (x^2 + y^2, 0) will move around the origin counterclockwise by how many rounds?'
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'Given an ellipse C with equation \\\frac{x^{2}}{3}+y^{2}=1\ and two fixed points \\(\\mathrm{A}(0,-1), \\mathrm{P}\\left(\\frac{3}{2}, \\frac{1}{2}\\right)\\). Let \\\mathrm{Q}\ be a moving point on the ellipse C. Determine the coordinates of point \\\mathrm{Q}\ and the area of \\\triangle \\mathrm{APQ}\ when the area is maximized.'
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'Graph the region depicted by the following inequalities:\n(1) \ \\frac{x^{2}}{9}+\\frac{y^{2}}{4}<1 \\n(2) \ \\frac{x^{2}}{9}-\\frac{y^{2}}{4} \\geqq 1 \'
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'Consider a triangular prism with vertices A(1,1,0), B(1,-1,0), C(-1,-1,0), D(-1,1,0), E(1,0,1), F(-1,0,1), and a right circular cone with the origin as the center and a circle on the xy-plane as the base. Find the minimum volume of such a cone and the radius r of its base when this minimum volume occurs.'
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'For polar coordinates, find the equations of the following circle and line: (1) A circle with center at point A(3, π/3) and radius of 2. (2) A line passing through point A(2, π/4) and perpendicular to OA (where O is the pole).'
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'(1) Ellipse \\\frac{x^{2}}{3}+\\frac{y^{2}}{2}=1\\\n(2) Hyperbola \\(\\frac{x^{2}}{4}-\\frac{(y-1)^{2}}{5}=1\\)'
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'On the plane in coordinate space, find the area of region 57 where the distance to the x-axis and y-axis is both less than or equal to 1.'
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'In the xy-plane, let the line passing through the point (1,2) with slope t be denoted as l. Also, let P be the point of intersection between the line perpendicular to l passing through the origin and l.\n(1) Express the coordinates of point P in terms of t.\n(2) Determine the value of a such that the trajectory of point P is a second-degree curve 2x^2-ay=0 (a≠0) and shares only 3 points. Also, find the coordinates of these 3 shared points.'
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'Find the locus of points that satisfy the following conditions: the ratio of the distance from point F and the distance from the fixed line l is e:1. Here, e>1, F is (c, 0), and l is the y-axis (x=0).'
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'Please provide the formula to calculate the area S of triangle OAB.'
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'In comprehensive mathematics II 11, let a>0. Let O be the origin in the coordinate plane, and let point P(1,3) be the point on the ellipse ax²+y²/2a=1 from which 2 tangents are drawn to the ellipse. Let the points where the tangents touch the ellipse be Q and R. Points Q and R lie on the line. Furthermore, let M be the midpoint of QR. Point M lies on the line y=ax. Moreover, when considering the areas of triangles PQR and OQR as S₁ and S₂, respectively, the ratio S₁/S₂ takes its minimum value when a=U. [Similar to Ritsumeikan University] => Original Exercise 47, 54: Let Q(x₁, y₁) and R(x₂, y₂). Then, the equations of the tangents at points Q and R are [ax₁x+y₁/2ay=1, ax₂x+y₂/2ay=1], both passing through point P(1,3) which implies [ax₁+3/2ay₁=1, ax₂+3/2ay₂=1]. This indicates that the line aax+3/2ay=1 goes through the points Q, R. Since Q and R are different points, ① represents the equation of the line QR. Rearranging (1) gives y=2a/3(1-ax). Substituting this into the equation of the ellipse results in ax²+1/2a*4a²/9(1-ax)²=1, which simplifies to a(2a²+9)x²-4a²x+2a-9=0. Since, x₁ and x₂ are the two solutions of this quadratic equation, then based on the relationship between the roots and coefficients, x₁+x₂=4a²/a(2a²+9)=4a/2a²+9. Let M(X,Y), then X=(x₁+x₂)/2=2a/2a²+9. Also, Y=2a/3(1-aX)=2a/3(1-a*2a/2a²+9)=6a/2a²+9. From (3) and (4), Y=3X, thus point M lies on the line y=3x. Since point P also lies on the line y=3x, points O, M, P are collinear. Therefore, S₁/S₂=PM/OM=OP-OM/OM=1-2a/2a²+9/2a/2a²+9=2a²+9-2a/2a=a+9/2a-1. Since a>0, we can conclude using the principle of arithmetic mean greater than or equal to geometric mean that S₁/S₂≥2√(a*9/2a)-1=3√2-1. On the ellipse x²/p²+y²/q²=1, the equation of the tangent at point (α,β) can be written as [αx/p²+βy/q²=1]. A line passing through two distinct points exists uniquely. By multiplying by 9 and simplifying, M is the midpoint of QR, and M lies on the line QR. From (4), Y=3*2a/2a²+9. Since triangles PQR and OQR share the same base QR, the ratio of S₁ to S₂ is equal to the ratio of their heights. When p>0 and q>0, (p+q)/2≥√(pq) holds true. Equality holds when p=q.'
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'Understand the basic concepts about the standard form of an ellipse ().'
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'Let d be a positive constant. Consider the ellipse E determined by the sum of distances from points A(-d, 0) and B(d, 0) to a point P being 4d.'
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'Practice If a circle with center at point A(a,-a) has exactly 2 intersection points with the hyperbola C: xy=1, express the radius r of circle A in terms of a.'
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'Prove that the sum of distances from the origin to the points and , where the tangent line at point on the curve (not on the axes) intersects the -axis and -axis, is constant.'
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'In triangle ABC, where AB=2, AC=1, ∠A=x, and f(x)=BC. (1) Express f(x) as a function of x. (2) Let R be the radius of the circumcircle of triangle ABC, express \x0crac{d}{dx} f(x) in terms of R. (3) Find the maximum value of \x0crac{d}{dx} f(x). [Nagaoka Institute of Technology]'
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'When the point P(x, y) moves along the circumference of the circle x^2 + y^2 = r^2, what kind of curve does the point Q represented by coordinates (y^2 - x^2, 2xy) move on?'
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'Find the coordinates of the midpoint and the length of the chord formed by the intersection of the line y = 4x + 1 and the ellipse 4x^2 + y^2 = 4.'
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'On the coordinate plane, on the curve C: , the point P(1, ) is taken.'
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'Find the polar equation of a circle with center at the pole O and radius a.'
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'Find the polar equation of a circle with center at (a, 0) and radius a.'
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'Find the equation of an ellipse with foci at (2,0),(-2,0) such that the sum of distances from these two points is 6'
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'Find the equation of the line that is tangent to the curve y = √(25-x^2) with a slope of -⅗.'
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'Let be a hyperbola on which there is a point . Given that . Let the tangent to at point intersect the lines and at points and respectively. Prove that the circle with diameter passes through the two foci of .'
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'In the figure to the right, when a square ABDE and a square ACFG are constructed outside of triangle ABC, answer the following questions.'
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'Find the equation of a parabola with the vertex at the origin, the focus on the x-axis, and passing through the point (9, -6).'
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'Draw the outline of the curve \\( \\left\\{\egin{array}{l}x=\\cos \\theta \\\\ y=\\sin 2 \\theta\\end{array}(-\\pi \\leqq \\theta \\leqq \\pi)\\right. \\) (without considering concavity or convexity).'
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'In the plane, there is a circle C with center at the origin O and radius 5. Let a circle C_n with radius n rotate around C without slipping while being internally tangent to C. There is a point P_n on circle C_n. Initially, when the center O_n of circle C_n is at (5-n, 0) and point P_n is at (5,0), assume that the center of circle C_n rotates n times counter-clockwise inside circle C and returns to its original position. Let S_n be the point of contact between circle C and circle C_n, and let t be the angle that segment OS_n makes with the positive direction of the x-axis. (1) Express the coordinates of point P_n in terms of t and n. (2) Show that the curves described by point P_2 and point P_3 are the same.'
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'Find the equation of the ellipse such that (2) ellipse \\\frac{x^{2}}{3}+\\frac{y^{2}}{5}=1\ with coincident foci and a minor axis length of 4'
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'Prove that, in the external construction of triangle ABC, when squares ABDE and ACFG are constructed, BG=CE, and BG is perpendicular to CE.'
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'A parabola with its axis as the x-axis and tangent to the line y=x at (3,3) exists. The coordinates of the focus of this parabola are (x, y) = (39, ?), and the equation is of the form y = ?.'
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'(1) The part of the hyperbola excluding the point (-2, 0)\n(2) The part of the parabola where '
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'Find the locus of the center of a circle passing through the point F(4,0) and tangent to the line l: x=-4.'
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'Prove that when the normal to the ellipse C at point P intersects the x-axis at point Q, PF1 / PF2 = QF1 / QF2.'
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'Let a be a positive constant. Find the value of a such that two lines passing through the point (1, a) and tangent to the hyperbola x²-4y²=2 are perpendicular.'
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'Consider the ellipse and the hyperbola . Prove that if the foci of and coincide, then the tangents to and at their intersection points are perpendicular to each other.'
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'Let a > 0. Fix one end of a string of length 2πa at a point A on the circumference of a circle with radius a, and wind it around the circle. Find the length of the curve traced by the other end P of the string as it is unwound from the circle.'
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'Find all the values of z that make the two lines OA and OB intersect perpendicularly.'
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"There are circles C1: x^2+y^2-2x=0 and C2: x^2+y^2-x=0 on the xy plane. For a moving point P on circle C1, excluding the origin O, let Q be the intersection point of the line OP and circle C2, excluding O, and let Q' be the point symmetric to Q with respect to the x-axis. Find the equation representing the trajectory of the midpoint M of line segment PQ', and illustrate its general shape."
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'When a circle with center (3,3) touches the hyperbola xy=1 at two points, find the x-coordinate of the point of tangency.'
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'What is the standard form equation of a hyperbola? Also, please provide the coordinates of its foci.'
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'Find the locus of point P that satisfies the following conditions:\n(1) The ratio of distances from point F(1,0) and the line x=3 to point P is 1:sqrt(3)\n(2) The ratio of distances from point F(3,1) and the line x=4/3 to point P is 3:2'
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'Translate the given text into multiple languages.'
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'Please state the condition for the three points A(α), B(β), and C(γ) to be collinear.'
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'On the coordinate plane, find the coordinates of point P that rotates point A(2,1) by π/4. The coordinates of the point obtained by rotating point A by π/4 around point A are (1−√2,−2+2√2). Find the coordinates of point P.'
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'There is an ellipse E on the xy-plane with the origin as its center. Its major axis lies on the x-axis. With a length of 2a, and the minor axis with a length of 2b (a>b). What is the equation of ellipse E?'
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'The line passing through the two points of tangency of the two tangent lines drawn from a point P(x_0, y_0) on the hyperbola x^2-y^2=1 to the circle x^2+y^2=1 is denoted as line l. Where y_0 is not equal to 0.'
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'A line passing through the focus of the parabola intersects the parabola at points and . If the line segment is internally divided at point in the ratio , determine the slope of the line .'
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'A circle S_1 with radius 1 is inscribed with an equilateral triangle T_1. Circle S_2 is inscribed in T_1, and a square U_1 is inscribed in S_2. Furthermore, circle S_3 is inscribed in U_1, triangle T_2 in S_3, circle S_4 in T_2, and square U_2 in S_4, and so on, creating sequences of circles S_1, S_2, S_3,..., triangles T_1, T_2, T_3,..., squares U_1, U_2, U_3,...'
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'Let P(x1, y1) be a point on the hyperbola C: x^2/a^2 - y^2/b^2 = 1 (a>0, b>0), where x1>a. Let Q and R be the points of intersection of the tangent to C at point P and the lines x=a and x=-a, respectively. Prove that the circle with QR as diameter passes through the two foci of C.'
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'From the point P(4, t) (t≥0) on the line x=4, two tangents drawn to the ellipse E: x^{2}+4y^{2}=4 form an acute angle θ. Find (1) an expression for tan θ in terms of t. (2) Determine the value of t when θ is maximized. [Tokyo University of Science]'
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'Find the equations of the common tangents of the parabola y = 3/4 x^2 and the ellipse x^2 + y^2/4 = 1.'
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'Find the coordinates of point P on the hyperbola that minimizes the distance to point A(0,2), and the minimum distance at that point.'
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'Points A and B are on the lines y=x and y=-x, respectively. Find the locus of point P that divides segment AB in a 2:1 ratio when the area of triangle OAB is k (k is a constant). Here, O is the origin.'
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'A circle C: x^{2}+y^{2}=9 is inside which a circle D with radius 1 rolls without slipping. At time t, circle D is tangent to C at the point (3cos t, 3sin t).'
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'Find the equation of the tangent line drawn from point (-1,3) to the ellipse \ \\frac{x^{2}}{12}+\\frac{y^{2}}{4}=1 \.'
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'(1) \\left(t \\sqrt{1+\\sin ^{2} t}, 0\\right) (2) \\left(0, \\frac{t}{\\sin t}\\left(1+\\sin ^{2} t+\\sqrt{1+\\sin ^{2} t}\\right)\\right) (3) point (0,2)'
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'Proof: Let P and Q be the two points of intersection other than the origin O of two lines that are perpendicular at the origin and the parabola y^2=4px(p>0). Then, the line segment PQ always passes through a fixed point on the x-axis.'
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'Find the equation of the plane passing through the origin O and perpendicular to the y-axis.'
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'Draw a perpendicular line from point A(-1,2) to the line x-3y+2=0, and let H be the intersection point of this line with the given line.'
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'Find the equation of a parabola with focus at (0, -1) and directrix as the line y=1.'
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'Find the locus of the point P which divides the line segment AB in the ratio 1:2, where the endpoints A and B move along the x-axis and y-axis respectively, for a line segment AB of length 3. If the coordinates of points A and B are (s, 0) and (0, t) respectively, then AB² = 3² implies that s² + t² = 3² (1). Let the coordinates of point P be (x, y), and P divides AB in the ratio 1:2, hence x = 2s, y = -t. Therefore, s = 1/2 x, t = -y. Substituting these into (1) gives (1/2x)² + (-y)² = 3², which simplifies to x²/6² + y²/3² = 1. Thus, the locus of point P is an ellipse x²/36 + y²/9 = 1.'
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'Translate the given text into multiple languages.'
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'A hyperbola with asymptotes that intersect at right angles is called a rectangular hyperbola. Find the equation of the rectangular hyperbola with center at the origin and one focus at (0, 4).'
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'The set of points P(z) that satisfy the equation |z-α| = r(r>0) is a circle with center at point A(α) and radius r. Furthermore, the set of points P(z) that satisfy the inequality |z-α| ≤ r(r>0) is the circle with center at point A and radius r including its circumference and interior.'
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'When k=\\frac{2 \\sqrt{10}}{3}, the coordinates of the midpoint are \\left(-\\frac{3 \\sqrt{10}}{10}, \\frac{\\sqrt{10}}{15}\\right). When k=-\\frac{2 \\sqrt{10}}{3}, the coordinates of the midpoint are \\left(\\frac{3 \\sqrt{10}}{10},-\\frac{\\sqrt{10}}{15}\\right)'
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'Let the quadrilateral ABPC be inscribed in a circle.'
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"(1) Find the locus of points on a plane where the difference in distance from two fixed points and is non-zero and constant."
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'Let a and h be positive constants. The locus of point P, where the distance from the origin (0,0) and the distance from the line x = -a are in the ratio h:1, is denoted as C.'
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'When the length of one side of triangle ABC is 1, let triangle ABC be an equilateral triangle. Let point P on the plane containing triangle ABC move such that 34 AP⋅BP - BP⋅CP + CP⋅AP = 0 holds. Find the shape formed by P.'
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'Find the vector equation for the point of contact between the circle and the line'
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'In triangle OAB with OA=4, OB=5, and AB=6, let H be the circumcenter of triangle OAB. Express OH in terms of OA and OB.'
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'We have learned about the conditions for a quadrilateral to be inscribed in a circle, namely the conditions for quadrilateral ABCD to be inscribed in a circle. Here, in the complex plane, given different 4 points A(α), B(β), C(γ), D(δ), when no 3 points are collinear, we consider the condition for these 4 points to lie on a single circle.'
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'Find the equation of a circle passing through the three points O(0,0), A(2,1), B(1,2) on the xy plane.'
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'Study the five centers of a triangle and position vectors'
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'Example 32 | Vector equation of a tangent to a circle (1) Show that the vector equation of a tangent to a circle C with center C(c) and radius r at a point P₀(p₀) on C is (p₀-c)·(p-c)=r²(r>0). (2) Prove that the equation of the tangent at point (x₀, y₀) on the circle x²+y²=r²(r>0) is x₀x+y₀y=r² using vectors.'
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'Find the minimum value of the area S(t) of triangle ABC, where points A, B are (2,2) and point C is (t, -1, 4).'
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'Find the polar equations of the following circle and line. (3) The line passing through point A(√3, π/6) and perpendicular to OA.'
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'Hyperbola \\( \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0) \\quad[ \\) standard form]\\nThe center is the origin.\\nThe curve is symmetric with respect to the \ x \ axis, \ y \ axis, and the origin.\\nThe foci are \\( \\mathrm{F}(c, 0), \\mathrm{F}^{\\prime}(-c, 0) \\quad c=\\sqrt{a^{2}+b^{2}} \\)\\nThe asymptotes are the lines \ \\frac{x}{a}-\\frac{y}{b}=0, \\frac{x}{a}+\\frac{y}{b}=0 \\\nAt any point \ \\mathrm{P} \ on the curve, \\\left|\\mathrm{PF}-\\mathrm{PF}^{\\prime}\\right|=2a \'
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'Find the arc length and area of a sector with radius r and central angle θ (in radians).'
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'Let \ \\triangle \\mathrm{ABC} \ be an equilateral triangle with side length 1. When a point \ \\mathrm{P} \ on the plane containing \ \\triangle \\mathrm{ABC} \ moves such that \ \\overrightarrow{\\mathrm{AP}} \\cdot \\overrightarrow{\\mathrm{BP}} - \\overrightarrow{\\mathrm{BP}} \\cdot \\overrightarrow{\\mathrm{CP}} + \\overrightarrow{\\mathrm{CP}} \\cdot \\overrightarrow{\\mathrm{AP}} = 0 \, find the shape traced by \ \\mathrm{P} \. [Saitama University]'
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'Practice: Find the locus of the midpoint M of the line segment PQ when a line passing through the point (2,0) intersects the ellipse x^2+4y^2=1 at two distinct points P and Q. [Similar to Shizuoka University]'
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'Translate the given question into multiple languages.'
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'There is a moving point P on the circumference of a circle with center O and a line segment AB of length 2r as diameter. Let the area of triangle ABP be S1 and the area of sector OPB be S2. Answer the following questions:'
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"Let's call the point obtained by moving a distance u (0 ≤ u ≤ √3) in the direction from the origin O to the point F on the segment OF as U. Find the radius r of the circle in the section when cutting the plane perpendicular to line OF at point U as a function of u.\nHere, let DS and ET be the perpendiculars from points D and E to the line OF, respectively."
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'Find the magnitude of angle LMN in three-dimensional space given the points L(2,1,0), M(1,2,0), N(2,2,1).'
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'Let F(p, 0) (p ≠ 0) be the focus and line ℓ: x=-p be the directrix of the parabola. Let P(x, y) be a point on the parabola, and let PH be the perpendicular from point P to line ℓ.'
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'Let the plane defined by points A(1, -1, 0), B(3, 1, 2), and C(3, 3, 0) be denoted as α. Determine the relationship satisfied by x, y, z when point P(x, y, z) lies on α.'
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'Practice finding the maximum value of the acute angle Theta formed by the two tangent lines drawn from point P(5, t) on the line x=5 to the ellipse x^2/5+y^2=1. Also find the value of t that gives this maximum angle.'
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'Important Example 115 Rotational Movement of Quadratic Curves'
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'Based on the given text, explain the fundamentals of analytical geometry.'
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'In \ \\triangle ABC \, with side lengths \ AB=3, BC=\\sqrt{7}, CA=2 \ and circumcenter O. Let \ \\overrightarrow{AB}=\\vec{b} \ and \ \\overrightarrow{AC}=\\vec{c} \, answer the following questions: (1) Find the dot product \ \\vec{b} \\cdot \\vec{c} \. (2) Express \ \\overrightarrow{AO} \ in terms of \ \\vec{b} \ and \ \\vec{c} \.'
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'From the point A(4,5), draw the perpendicular from point A to the line ℓ: x+2y-6=0 and let the intersection point be H. (1) Find the coordinates of point H using vectors. (2) Determine the length of the line segment AH.'
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'Find the area of triangle OAB when the coordinates of point A in polar coordinates are (3, π/6) and the coordinates of point B are (5, 5π/6).'
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'Important question 134: Parametric representation of hypocycloid\nLet a>2 and b. A circle C with radius b rotates without slipping while being tangent to the circle O with radius a centered at the origin O. When a fixed point P(x, y) on circle C is initially at a fixed point A(a, 0) on the circumference of circle O, and the rotation angle from the positive x-axis of the line segment connecting the center C of circle C and the origin O is θ, express the curve that P traces in terms of the parameter θ.'
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'When the endpoints A and B of a line segment AB of length 3 move along the x-axis and y-axis, respectively, find the locus of the point P which divides the line segment AB externally in the ratio 1:2.'
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'When the points O(0), A(α), B(β) are not collinear, define addition as C(α + β), what shape will quadrilateral OACB take?'
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'In triangle ABC with A=4, B=5, C=6, let H be the circumcenter of triangle ABC expressed in terms of A and B. Let A=a, B=b. The midpoints of sides A and B are denoted as M and N, respectively. Since triangle ABC is not a right triangle, H does not coincide with M or N. Since H is the circumcenter of triangle ABC, we have AH ⊥ MH and BH ⊥ NH, so OH=s𝑎+t𝑏, where s and t are real numbers. From AH ⊥ MH, we have A⋅MH=0, so a⋅(OH−OM)=0 gives (s−1/2)a+t𝑏=0, and b⋅(OH−ON)=0 gives b⋅{s𝑎+(t−1/2)b}=0. Therefore, (s−1/2)|a|2+t𝑎⋅𝑏=0. From BH ⊥ NH, B⋅NH=0 implies s𝑎⋅b+(t−1/2)|b|2=0.'
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'When the point P(x, y) moves along the circumference of the fixed circle x^2+y^2=r^2, what kind of curve does the point Q with coordinates (x^2-y^2, 2xy) move on?'
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'When the point P(x, y) moves along the circumference of the circle x^2 + y^2 = 4, and the point Q with coordinates (x^2 / 2 - y^2 + 3, 5/2xy - 1), what kind of curve does it move along?'
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'In an isosceles triangle ABC with AB = AC = 1, find the length of the base that maximizes the area of the circle inscribed in the triangle. (Tokyo University of Science)'
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'The vertices of a regular hexagon with side length 1 are labeled as A, B, C, D, E, F in both clockwise and counterclockwise directions, with AB denoting vector a and AF denoting vector b.'
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'The length of segment AB is 2, with point A moving on the x-axis and point B on the y-axis. In this case, extending segment 105AB, find the locus of point P such that BP=1.'
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'Which one of the following is the equation for the asymptotes of the hyperbola x^2/a^2 - y^2/b^2 = 1?'
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'Center (1,0); vertices (7,0), (-5,0); asymptotes y = \\frac{1}{3} x - \\frac{1}{3}, y = -\\frac{1}{3} x + \\frac{1}{3}'
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'Find the equation of an ellipse with center at the origin, major axis on the x-axis, minor axis on the y-axis, and passing through the points (-4,0) and (2, √3).'
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'Find the locus of points P where two tangents drawn from an external point P(a, b) to the ellipse are perpendicular.'
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'Example 95 | Equation of a Line on the Complex Plane (2)\nIn the complex plane, draw two tangents from point A(α) (|α|>1) to a circle with center at the origin O and radius 1. Let B and C be the two points of tangency with the circle, and let point P(z) lie on the line BC. Let β be the complex number representing point B. Show that ᾱz + αz̄ is constant regardless of the choice of points A and P, and determine its value.\n[Similar to Tokushima University]'
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'Find the equation of the line that is tangent to both curves y=-x^{2} and y=\\frac{1}{x}.'
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'Find the equation of an ellipse with foci at (2√2, 0) and (-2√2, 0) such that the sum of distances from the foci is 6.'
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'Find the equation of the tangent line drawn from point A(0,5) to the ellipse x^{2}+4 y^{2}=20.'
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'PRACTICE 70\nFor the following curves, find the equation of the tangent line at the point corresponding to the specified value of t in ().\n(1) \\( \\left\\{\egin{array}{l}x=2 t \\\\ y=3 t^{2}+1\\end{array} \\quad(t=1)\\right. \\)\n(2) \\( \\left\\{\egin{array}{l}x=\\cos 2 t \\\\ y=\\sin t+1\\end{array} \\quad\\left(t=-\\frac{\\pi}{6}\\right)\\right. \\)'
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'When two linearly independent vectors and are defined in a plane, any point can be uniquely represented as where are real numbers. In this case, the pair of real numbers is called oblique coordinates. Please solve the following problem using these oblique coordinates:\n\nIf the point lies on the line in the Cartesian coordinate plane, how is it represented in the oblique coordinate plane?'
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'Find the area S of the triangle PQR with vertices P(2,8), Q(0,-2), R(6,4).'
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'Explain the existence range of points on a plane, particularly in the following geometric shapes: 1. Line AB 2. Triangle OAB 3. Parallelogram OACB'
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'Let O be the origin. As point P lies on plane α, it can be expressed as s, t, u are real numbers and ∠OP=sOA+tOB+uOC, s+t+u=1. Therefore, (x, y, z) =s(1,2,4)+t(-2,0,3)+u(4,5,-2) = (s-2t+4u, 2s+5u, 4s+3t-2u) which implies x=s-2t+4u, y=2s+5u, z=4s+3t-2u. Solving for s, t, u we get s=(1/39)(15x-8y+10z), t=(1/39)(-24x+18y-3z), u=(1/39)(-6x+11y-4z). Substituting into s+t+u=1 and simplifying we get 5x-7y-z+13=0. Another solution method is to consider the equation of two planes as ax+by+cz+d=0. Passing through point (1,2,4) gives a+2b+4c+d=0, passing through point (-2,0,3) gives -2a+3c+d=0, passing through point (4,5,-2) gives 4a+5b-2c+d=0. From (1), (2), (3) we can find a=-5c, b=7c, d=-13c, thus −5cx+7cy+cz-13c=0 assuming c≠0, simplifies to 5x-7y-z+13=0. When point P lies on plane α, it satisfies this equation, which is the required relationship.'
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'Concerning polar coordinates, find the polar equations for the following circle and lines.'
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'In space, there is a triangle ABC with vertices A(5,0,1), B(4,2,0), and C(0,1,5). (1) Find the lengths of the line segments AB, BC, and CA. (2) Find the area S of the triangle ABC.'
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'Translate the given text into multiple languages.'
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'Find the polar equations of the following circle and line. \n(1) Circle with center O and point A(4, π/3) as the endpoints of the diameter'
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'A circle, ellipse, hyperbola, and parabola are each represented by the following second-degree equations in x and y.'
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'Example 137 Polar coordinates and trajectories Let the polar coordinates of point A be (2,0), and let Q be any point on the circumference of the circle C with diameter OA connecting the pole O and point A. Let the tangent of circle C at point Q, drop a perpendicular OP from the pole O to point P, and let the polar coordinates of point P be (r, θ). Find the polar equation of the trajectory of point P. Where 0 ≤ θ < π.'
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'(1) Find the equations of the lines passing through the point A(-2,3) that are parallel and perpendicular to the line ℓ: 5x+4y-20=0.'
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'Find the locus of the centers of the circles that are tangent to both the circle with center A(2,0) and radius 1 and the line x=-1, and do not contain point A.'
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'Problem 105 Ellipse and Trajectory\nThere is a line segment AB with a constant length l (>0), where point A lies on the x-axis and point B moves along the y-axis. Find the locus of the point P that divides the line segment AB internally in the ratio m: n. Here, m > 0, n > 0, and m ≠ n.'
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'Find the polar coordinates (r, θ) [0 ≤ θ < 2π] for the following Cartesian points C, D.'
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'Practice (1) Find the eccentricity of an ellipse with a major axis of length 4 and a minor axis of length 2. Also, find the polar equation of the ellipse with a focus at 138O and the semilatus rectum perpendicular to the transverse axis being the initial line. (2) Find the polar equation of a line passing through point B on the ellipse from (1) (with OB=2) and perpendicular to OB. Assume B is located above the line θ=0.'
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'Example 141 Parametric representation of an epitrochoid'
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'On the coordinate plane with the origin at O (0, 0), on the curve C: x ^ 2 / 4 + y ^ 2 = 1, a point P (1, √3/2) is taken.'
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'Explain the definition and properties of an ellipse.'
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'1. Perpendicular bisector of the line segment connecting point 0 and point 1\n2. Circle with center at point 1/2 and radius 1/2. Excluding point 1'
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'Prove that if the line segments PO and QO passing through the ends of the chord PQ of the parabola y^2 = 4px (p>0) and the origin O are perpendicular, then the chord PQ passes through a fixed point.'
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'Find the locus of points P that satisfy the following conditions:\n(1) The ratio of the distance from point F(1,0) to the line x=4 to the distance from point P is 1:2\n(2) The ratio of the distance from point F(1,0) to the line x=4 to the distance from point P is 2:1'
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'Equation of a plane parallel to the coordinate plane: passing through point P(a, b, c), a plane parallel to the yz-plane equation is... x=a, a plane parallel to the zx-plane equation is... y=b, and a plane parallel to the xy-plane equation is... z=c. In particular, the equations of the xy-plane, yz-plane, and zx-plane are z=0, x=0, and y=0 respectively.'
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'For a natural number n, define P_n and P_{n+1} as follows: Let Q_n be the intersection point of the tangent line at point P_n on curve C and the x-axis, and let P_{n+1} be the intersection point of the line passing through Q_n and perpendicular to the x-axis with curve C. Find the area of the region bounded by C and the line segments P_nQ_n, Q_nP_{n+1}, denoted as S_n.'
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'(1) Given a point z on a circle with center at the origin and radius 3, find the locus of point w.'
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'Prove that the product of the lengths of the line segments \ \\mathrm{PQ} \ and \ \\mathrm{PR} \ drawn as perpendiculars from an arbitrary point \ \\mathrm{P} \ on the hyperbola \\( \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0) \\) to the two asymptotes is constant.'
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'(1) Plot the graph with vertices at (2,0), (-2,0), foci at (2√2, 0), (-2√2, 0), and asymptotes at y=±x. (2) Plot the graph with vertices at (0,5), (0,-5), foci at (0, √34), (0,-√34), and asymptotes at y=±5/3x.'
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'A circle C: x^2 + y^2 = 9 on the coordinate plane, inside which a circle D with radius 1 rolls without slipping. At time t, D is tangent to C at the point (3 cos t, 3 sin t). At time t = 0, point P on D was at (3,0). Find the coordinates (x(t), y(t)) of point P at time t, where 0 ≤ t ≤ 2/3π.'
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'(1) Set the values of a, d, f to be a=2, d=-10, f=0, and then, when certain values were assigned to b, c, an ellipse as shown in Figure 1 was obtained. In this case, the most suitable combination of values for b, c is which of the following (choose from 0 to 7) is A.'
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'Find the coordinates of the midpoint of the chord formed by the intersection of the line y=x+2 and the ellipse x^{2}+3 y^{2}=15, as well as its length.'
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'Find the minimum distance between a point P on the parabola y^2=4x and the fixed point A(a, 0). Here, a is a constant.'
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'Find the coordinates of the fourth vertex S of the parallelogram with vertices P(1,2), Q(3,-2), R(4,1).'
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'Consider a circle of radius r (r ≤ 1) with its center moving around the perimeter of a square with side length 4 in the plane. Find the area S(r) of the circle that intersects with the square.'
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'Find the equation of a sphere that is tangent to the zx-plane with center at (2, -3, 1).'
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'A hyperbola with perpendicular asymptotes is called a rectangular hyperbola. Find the equation of the rectangular hyperbola with the center at the origin and one focus at (0,4).'
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"For a positive real number t, let the two points on the plane be F(t, 0) and F'(3t, 0), and the sum of distances to these two points is 2√2t for a point P whose trajectory is C. Let the line y=x-1 be represented as l. (1) Find the range of values for t that result in C having two distinct intersection points with l. (2) Determine the maximum area of a triangle with vertices C, the two intersection points, and the origin O as t varies within the range found in (1). [Kumamoto University]"
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'(1) Find the equation of a hyperbola with foci at (0,5) and (0,-5), and a difference in distance from the foci of 8.'
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'Find the vertex, focus, and asymptotes of the following hyperbola. Also, sketch its rough shape.'
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'Given an ellipse \ \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \ and a point \ \\mathrm{P} \ outside the ellipse from which two tangents are drawn to the ellipse such that they are perpendicular, answer the following questions:\n(1) Find the coordinates of point \ \\mathrm{P} \ at which the two tangents become parallel to the \ x \ axis or \ y \ axis.\n(2) Determine the locus of point \ \\mathrm{P} \.'
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'Find the parametric representation of an ellipse.'
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'Find the locus of the centers of circles that are tangent to the circle with equation (x-3)^2 + y^2 = 1 and also tangent to the line x = -2.'
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'Find the polar equation of a circle with center O and radius a.'
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'Prove that the incenter P(z) of triangle OAB with vertices O(0), A(α), B(β) satisfies the equation z=|β|α+|α|β/|α|+|β|+|β-α.'
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'In the plane, there is a triangle \ \\triangle \\mathrm{OAB} \, where \ \\mathrm{OA}=5, \\mathrm{OB}=8, \\mathrm{AB}=7 \. Let \ s, t \ be real numbers, such that point \ \\mathrm{P} \ is defined as \ s \\overrightarrow{\\mathrm{OA}}+t \\overrightarrow{\\mathrm{OB}} \. (1) Find the area \ S \ of \ \\triangle \\mathrm{OAB} \. (2) If \ s \\geqq 0, t \\geqq 0, 1 \\leqq s+t \\leqq 2 \, then define the area of the region where point \ \\mathrm{P} \ lies as \ T \. Find the ratio of areas \ S: T \. [Similar to Josai University]'
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'In triangle OAB, with OA=1, OB=2, and ∠AOB=45°, let H be the orthocenter. If vector OA=a and vector OB=b, express vector OH in terms of a and b.'
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'Find the coordinates of point P that divides line segment AB internally in the ratio m:n.'
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'Let α=x+4i, β=6+6xi. When 2 points A(α), B(β) and the origin O are collinear, find the value of the real number x.'
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'Practice: In quadrilateral \\\mathrm{ABCD}\, let \\\mathrm{P}\ and \\\mathrm{Q}\ be the midpoints of sides \\\mathrm{AB}\ and \\\mathrm{CD}\, respectively. Let \\\mathrm{M}\ and \\\mathrm{N}\ be the midpoints of the diagonals \\\mathrm{AC}\ and \\\mathrm{BD}\.'
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'Determine the value of a so that the lines AB and AC are perpendicular.'
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'How to handle conditions that are located on the same plane?'
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'Find the coordinates of the centroid G of triangle ABC.'
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'Find the area enclosed by \\( (x^{2}-2)^{2}+y^{2}=4 \\).'
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'Let B120 be k>0 and b>0. When point P moves on the circumference of the circle x^{2}+y^{2}=a^{2}, obtaining point Q whose y-coordinate is scaled by b/a, denoted as C1 as the trajectory of Q. Let k be a constant, defining a curve C2 symmetric to C1 with respect to the line y=x+k.\n(1) Find the equation representing C1.\n(2) Find the equation representing C2.\n(3) Determine the range of values for k when the line y=x+k does not have any shared points with C2.'
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'Find the polar equation of a circle with center at \\( (a, 0) \\) and radius \ a \.'
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'Through the point A(3,0) and the line l perpendicular to the initial line, find the polar equation of the parabola with O as the focus and l as the directrix.'
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'Let the intersection point of the diagonals of this regular hexagon be O.'
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'Consider two curves . and have 2 intersection points in the range , with their respective coordinates as . Let the region enclosed by the two curves in the interval be denoted as , and its area be denoted as . Furthermore, within , the area of the part where is , and the area of the part where is .\n(1) Express in terms of .\n(2) Express in terms of .\n(3) Determine the value of so that .'
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'Problem on circumcenter and vector equality in Example 24\nLet P, Q, R be the feet of the altitudes from the circumcenter O of acute triangle ABC to lines BC, CA, AB respectively. Given that OP+2OQ+3OR=0.\n(1) Prove that 5OA+4OB+3OC=0.\n(2) Find the dot product OB · OC.\n(3) Find the measure of angle A.'
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'(1) Find the focus and directrix of the parabola . Also, sketch its general shape. (2) Find the equation of the parabola where the focus is at point and the directrix is the line $y=1.'
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'[1] The circle K with center C and radius r (where r is a positive constant) has the condition for point P to be on circle K as |CP⇀|=r, which leads to |𝗽⇀-𝐜⇀|=r and |𝗽⇀-𝐜⇀|^{2}=r^{2}. When 𝗽=(x, y) and 𝐜=(a, b), the equation of the circle is obtained as (x-a)^{2}+(y-b)^{2}=r^{2}.'
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"(2) Let the intersection point of diagonals AC and BD be O, ∠AOB=θ, AO=x, BO=y, then OC=p-x, OD=q-y, hence S= △AOB + △BOC + △COD + △DOA. Heron's formula can be used."
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'From a point P outside the circle O, draw two tangents to the circle, let the points of contact be S and T, and let H be the intersection of OP and ST. Also, let a line passing through point P (not coinciding with OP) intersect the circle O at two points A and B. (1) Prove that △POS ∼ △PSH. (2) Prove that the four points A, B, H, O lie on a common circle.'
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'In △ABC, when the lengths of the three sides are as follows, whether △ABC is an acute triangle, a right triangle, or an obtuse triangle.(1) a=6, b=4, c=3'
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'Ship A, traveling north at a speed of 800 m per minute, and ship B, traveling west at a speed of 600 m per minute, are converging with their routes intersecting at point O. Currently, A is 2 km south of O and B is 4 km east of O. When the two ships get closest, what is the distance between them in km?'
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'Explain the isosceles triangle and its properties.'
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'In the quadrilateral ABCD inscribed in a circle, with AB=√2, BC=4, CD=3√2, and ∠BCD=45 degrees, answer the following questions.\n(1) Find the length of side DA.\n(2) Find the area of quadrilateral ABCD.'
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'Special parallelograms include the following types: \n[1] Rectangle\n (A) Has four equal angles (definition).\n (B) Diagonals are of the same length.\n[2] Rhombus\n (C) Has four equal sides (definition).\n (D) Diagonals intersect at right angles.\n[3] Square\n Both a rectangle and a rhombus. Meets all of the criteria (A) to (D) above.'
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'The tangent line l of a circle passing through point A is perpendicular to radius OA. If the line l passing through point A on the circumference is perpendicular to radius OA, then l is the tangent of this circle. Also, the lengths of the two tangents drawn from a point outside the circle to the circle are equal.'
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'Example 104 | Radii of Circumscribed and Inscribed Circles'
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'In triangle ABC with 30°, find the following values. (1) When b=3, c=√2, A=45°, find a'
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'A triangle has side lengths of 1, 3, and x. (1) Find the range of possible values for x. (2) Find the range of x when the triangle is acute-angled.'
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'Please solve a problem related to the properties of shapes.'
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'Prove that the centroid of triangle ABC coincides with the centroid of triangle DEF where D, E, and F are the midpoints of the sides BC, CA, and AB respectively.'
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'Practice: Color the A, B, C, D in the right figure with 4 colored pencils. Each part of A, B, C, D is an equilateral triangle.'
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'Let the angle between two lines (1) and (2) and the x-axis be α and β (0° < α < 180°, 0° < β < 180°).'
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'The conditions to be a parallelogram are as follows:\n[1] Two pairs of opposite sides are equal.\n[2] Two pairs of opposite angles are equal.\n[3] The diagonals intersect at their respective midpoints.\n[4] One pair of opposite sides are parallel and equal in length.'
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"In mathematics A-251, draw the lines AA' and BB'. These two lines are common internal tangents of the two circles O and O'."
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'Let there be a regular tetrahedron OABC with edge length 1. Let M be the midpoint of edge OB, and let point P move along edge OC. If we denote the length of segment OP by t, express (1) AP^2 and PM^2 in terms of t. (2) If we denote angle PAM as theta, express cos theta in terms of t. (3) Express the area of triangle AMP in terms of t. (4) Determine the minimum value of the area of triangle AMP. [Niigata University]'
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'Practice\n103rd book p.201\nLet OH be the perpendicular line drawn from O to AB, where H is the midpoint of the side AB. Given OA=OB=a, and ∠AOB=360° ÷ 8=45°, the area of △OAB is (√2/4) * a^2\nBy the cosine rule, AB^2=a^2 + a^2 - 2a * a * cos 45° = (2 - √2)a^2\nIn △OAH, AH^2=a^2 - r^2\nSince AH= (1/2) * AB, then (1/4) AB^2=a^2 - r^2\nTherefore, 4(a^2 - r^2)=(2 - √2)a^2'
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'In △ABC, find the following: (2) Find angle A when a=7, b=8, c=5'
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'Describe the properties of a quadrilateral inscribed in a circle.'
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'In triangle ABC, when the lengths of the three sides are as follows, what type of triangle is ABC - acute-angled triangle, right-angled triangle, or obtuse-angled triangle? (2) a = 5, b = 13, c = 12'
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'Calculate the length of the perpendicular line drawn from point A to the plane BCDEF.'
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'In triangle ABC, with the side lengths as follows, whether triangle ABC is acute-angled, right-angled, or obtuse-angled. (3) a=10, b=9, c=12'
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'For the circle O shown on the right, answer the procedure for drawing the following tangents:\n(1) Tangent to circle O at point P on the circumference of the circle\n(2) Tangent drawn from an external point Q to circle O'
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'In triangle ABC, where AB=2 and AC=1. Let D be the intersection point of the angle bisector of ∠BAC and side BC. If AD=BD, find the area of triangle ABC.'
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'What is the shape of the triangle with vertices A(2, -1, 2), B(0, 2, 3), and C(3, -4, 0)?'
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'Verify that the lengths of the three sides of a triangle are , , . If they are correct, explain the reason.'
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'Four points P, T, O, S are on the same circumference of a circle.'
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'Example Problem 50 | Maximum and Minimum Problem (1)\n(1) Cut a rope of length l into two pieces. Let one length be x, and use this length x of rope to make a circle. Use the other piece of rope to make a square. Express the sum S of the areas of the circle and square in terms of x and l.\n(2) Express the minimum value of S and the corresponding x in terms of l.\n[Similar to Chuo University]'
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'Calculate the distance between two points on a number line.'
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'Practice: In acute triangle ABC, let H be the orthocenter, M be the midpoint of side BC, and N be the midpoint of segment AH. When 48, prove using the result from the above example problem that the length of segment MN is equal to the circumradius of triangle ABC.'
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'There is a triangle with side lengths 2 and 3, and one angle measuring 60 degrees. Find the length of the remaining side of this triangle.'
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'In the following way, the minimum (a, b) can also be determined using the coordinate plane. In the ab plane (*), a^{2}+b^{2} represents the square of the distance between point (a, b) and the origin, and a+b=27k represents a line as shown in the right figure as k increases with k=1,2, ... . The lattice points on these lines where the distance from the origin is minimized are, as shown on the right figure, at points (13,14) and (14,13) on the line a+b=27. In other words, a^{2}+b^{2} reaches its minimum when (a, b)=(13,14),(14,13). (*) The axis that takes the values of a is the a-axis, and the axis that takes the values of b is the b-axis, forming the coordinate plane.'
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'As shown in the diagram, there are 5 parallel lines intersected by 3 parallel lines, all equally spaced. (1) How many rectangles (including squares) are there in the figure enclosed by 4 of the 8 lines? (2) Out of the 15 intersection points of these parallel lines, how many triangles can be formed by selecting 3 points, with point A as one of the vertices?'
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'Please explain the relationship between the sides and angles of a triangle.'
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'The position relationship between a circle and a line is classified based on the distance d from the center C of a circle with radius r to the line ℓ as follows: [1] When d<r, they intersect (with 2 intersection points). [2] When d=r, they are tangent (share a point of contact). [3] When d>r, they are apart.'
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'For the regular tetrahedron ABCD with side length 6, point E satisfying 2BE=EC on side BC, and point M as the midpoint of side CD. [Osaka Kyoiku University]'
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What is the shape of a right triangle whose two legs have a sum of 16 that maximizes the area? Also, find the maximum value.
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Find the minimum length of the hypotenuse in a right triangle where the sum of the lengths of the two legs is 10.
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2. Points P, Q, and R are taken on the sides AB, BC, and CD respectively of a square ABCD with a side length of 8, such that AP=x, BQ=2x, and CR=x+4 (0<x<4). The areas of triangles PBQ and QCR expressed in terms of x are ア (square) and イ (square) respectively. Therefore, the area of triangle PQR takes the minimum value of エ (square) (square) when x=ウ (square).
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What kind of shape does the following parametric representation describe? (1) (2) (3) x=rac{4}{\cos heta}+2, y=3 an heta-1
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The equation of the plane passing through point A(a, 0, 0) and parallel to the yz-plane is
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Find the equation of the tangent line at the point \( (\sqrt{2}, 1) \) on the ellipse .
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There is a circle centered at point \( \mathrm{A}(a, 0) \) with a radius of . Consider any point on this circle and form a square with the line segment (connecting the pole and point ) as one side. Find the polar equation of the locus of point .
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Show that the two tangents drawn from the point \( (2,1) \) to the ellipse x^{2}+rac{y^{2}}{4}=1 are perpendicular.
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Find the locus of point P such that the ratio of its distance from point F(0,1) to its distance from the line ℓ: y=-1 is as follows: (1) 1:1 (2) 1:2 (3) 2:1.
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Find the equation of the ellipse that satisfies the following conditions: (1) The foci are at (3,0) and (-3,0), and the difference between the lengths of the major and minor axes is 2. (2) The center is at the origin, the major axis is along the y-axis, the minor axis has a length of 8, and it passes through the point (12/5, 4).
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107 (1) Parabola (2) Ellipse \( rac{3}{4} x^{2}+rac{9}{16}\left(y-rac{5}{3}
ight)^{2}=1 \)(3) Hyperbola \( rac{3}{16} x^{2}-rac{9}{16}\left(y+rac{5}{3}
ight)^{2}=-1 \)
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A parabola \( C: y^2 = 4px (p>0) \) has its focus . Two chords and intersect perpendicularly and pass through .
(1) Find the polar equation of the parabola with as the pole and the positive portion of the x-axis as the polar axis.
(2) Show that is constant.
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What type of curve does the following equation represent? If it is an ellipse, find the center and foci; if it is a hyperbola, find the vertices, foci, and asymptotes; if it is a parabola, find the vertex, focus, and directrix. (1) (2) (3)
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For the ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>0, b>0) \) and its vertices \( \mathrm{A}(a, 0), \mathrm{B}(0, b) \), find the coordinates of the point in the first quadrant that maximizes the area of the quadrilateral . Also, find the value of at that point. Note that O is the origin.
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What type of curve does the following equation represent? If it is an ellipse, find the center and foci; if it is a hyperbola, find the vertices, foci, and asymptotes; if it is a parabola, find the vertex, focus, and directrix.
(1)
(2)
(3)
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Given the parabola \( y^{2}=4 p x(p
eq 0) \) with its focus at , and a line passing through the focus that intersects the parabola at two points , show that the product of the -coordinates of points and is constant.
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Express the condition for the line to be tangent to the ellipse x^{2} + rac{y^{2}}{4} = 1 using m and n.
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Find the polar coordinates of the following points in Cartesian coordinates: P(2, 2), Q(1, -√3), R(-√3, 3), S(-2, 0). The range for the angle θ should be 0 ≤ θ < 2π.
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From point P(1,3), draw a perpendicular line to the line ℓ: 2x-3y+4=0, with the intersection point H. (1) Using vectors, find the coordinates of point H. (2) Calculate the distance between point P and the line ℓ.
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On the coordinate plane, take point \( \mathrm{A}(2,0) \), and on the circumference of a circle with center at the origin and radius 2, take points , so that points successively become the vertices of a regular hexagon. Assume that B is in the first quadrant.
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When the point moves on a circle with a radius of 1 centered at the origin , what shape will the point described by the following equations draw?
(1)
(2)
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Advanced Example 38
In , given , , and , let the orthocenter be . If and , express in terms of and .
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Equation of a line drawn from a point not on the conic section
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Please explain the properties of the following parabola. x^2=4py (p≠0)
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Find the locus of the center \( \mathrm{P}(x, y) \) of a circle that is tangent to the line and passes through the point \( \mathrm{A}(-3,0) \).
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Find the polar equation of the curve for which the ratio of the distance from point P to the pole O and the distance to the line l is 1:2 in the given example.
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Find the equation of the ellipse that meets the following conditions. (1) It has two foci at (,(-) and the sum of the distances from any point on the ellipse to the foci is 8. (2) It has two foci at (,() and the sum of the distances from any point on the ellipse to the foci is 12.
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Find the equation of the ellipse that satisfies the following conditions: (1) The foci are at the points (,(-) and the sum of the distances from the foci is 8. (2) The foci are at the points (,() and the sum of the distances from the foci is 12.
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If we dilate the circle in the direction by a factor of 2, what curve do we get?
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92 (1) Right isosceles triangle with ∠O = π/2 (2) Right triangle with ∠O = π/2, ∠A = π/3, ∠B = π/6
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Example 2. The shape represented by the equation .
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Foci of an ellipse, lengths of the major and minor axes
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Find the equations of the following ellipses.
(1) rac{x^{2}}{5}+y^{2}=1
(2) rac{x^{2}}{4}+rac{y^{2}}{9}=1
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17 (1) \( \mathrm{M}(3,3,1), \mathrm{N}(2,3,3) \), area
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By considering the intersection of the parabola and the line , express this parabola using as a parameter.
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Find the equation of the hyperbola that meets the following conditions. (1) The foci are at \( (3 \sqrt{2}, 0),(-3 \sqrt{2}, 0) \), and the difference in distance from the foci is 6. (2) The foci are at \( (0, \sqrt{26}),(0, -\sqrt{26}) \), and the difference in distance from the foci is .
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Let TR be a non-zero constant. By considering the intersection of the parabola and the line , parametrize this parabola using as a parameter.
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Find the foci and asymptotes of the following 99 ellipses:
(1) Two points \( (\sqrt{29}, 0),(-\sqrt{29}, 0) \); Two lines y=rac{2}{5} x, y=-rac{2}{5} x ; Diagram omitted
(2) Two points \( (2\sqrt{2}, 0),(-2\sqrt{2}, 0) \); Two lines ; Diagram omitted
(3) Two points \( (0, \sqrt{34}),(0,-\sqrt{34}) \); Two lines y=rac{5}{3} x, y=-rac{5}{3} x ; Diagram omitted
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Find the value of the real number when points \( \mathrm{A}(lpha) \), \( \mathrm{B}(eta) \), and the origin are collinear.
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Chapter 4 Curves and Equations-105
EX In the coordinate plane, let the curve represented by the polar equation be , and let the points on with polar coordinates \( { }^{4} 51\left(\sqrt{2}, rac{\pi}{4}
ight),(2,0) \) be and respectively. Also, let the straight line passing through A and B be called , and let the circle centered at A with the line segment as its radius be called .
(1) Determine the polar equation of the straight line .
(2) Determine the polar equation of the circle .
[Kanazawa Institute of Technology]
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Find the equation of the hyperbola that satisfies the following conditions: ① The vertices are at (1,0) and (-1,0), and the asymptotes are y=3x and y=-3x; ② The foci are at F(6,0) and F'(-6,0), and one vertex is at the point (2√5, 0); ③ The difference in distances from a point on the hyperbola to the two foci F(0, 5) and F'(0, -5) is 8.
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Find the focus and directrix of the following parabolas, and sketch their shapes.
(ア)
(イ)
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By transforming the equation:
25(x^{2}+4x+2^{2})-25 * 2^{2}-4(y^{2}+6y+3^{2})+4 * 3^{2}-36 = 0
Therefore: 25(x+2)^{2}-4(y+3)^{2}=100
That is:
\(rac{(x+2)^{2}}{4}-rac{(y+3)^{2}}{25}=1\)
Do the following conic section and straight line have any common points? If so, determine whether they are intersection points or tangency points, and find the coordinates of those points.
(1) 4x^{2} + 9 y^{2} = 36 and 2x - 3y = 0
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Find the equation of the hyperbola that satisfies the following conditions: (1) The vertices are (1,0) and (-1,0), and the asymptotes are y=3x and y=-3x (2) The foci are at (6,0) and (-6,0), with one vertex at (2√5, 0) (3) The difference in distances from any point on the hyperbola to the foci F(0,5) and F' (0,-5) is 8.
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83 (1) A circle with a radius of 4 centered at point 1-i (2) A circle with a radius of 2 centered at point 1
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Similar to the example above, what kind of curve does the point \( (x, y) \) satisfy, represented by x=rac{1-t^{2}}{1+t^{2}}, y=rac{4 t}{1+t^{2}} where is a parameter?
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If the circle is scaled by a factor of in the axis direction with the axis as the base, what kind of curve will it become?
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Find the equation of the tangent line drawn from the point \( (1,3) \) to the ellipse rac{x^{2}}{12}+rac{y^{2}}{4}=1 .
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Find the polar equations of the following circles in polar coordinates with the origin O:
(1) A circle with center at the origin O and radius 3
(2) A circle with center at point A, where A has polar coordinates (4,0), and radius 4
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What kind of figures are represented by the following equations in and ?
(1)
(2)
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Given a circle with center at point \( \mathrm{A}(a, 0) \) and radius , consider any point on the circle and the segment connecting point with the pole . Construct a square with as one of its 1261 edges. Find the polar equation of the trajectory of point .
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What is the geometric figure represented by the set of points that satisfy the following equations in the complex plane?
(1)
(2)
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Find the equation of a circle in the coordinate plane. The center has coordinates (a, b) and the radius is r.
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The equation of the plane passing through point C(0,0,c) and parallel to the xy-plane: z=c
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Please explain the properties of the following ellipse. x^2/a^2 + y^2/b^2 = 1 (a > b > 0)
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Find the equation of the ellipse that satisfies the following conditions. (1) It has foci at (2,0) and (-2,0), and the sum of distances from any point on the ellipse to these foci is ; (2) It has foci at (,() and the sum of distances from any point on the ellipse to these foci is 6.
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Given a regular tetrahedron OABC with edge length 1, let P and Q be the midpoints of edges OA and OB, respectively. Let R be the point dividing edge OC in the ratio 3:2. Find the centroid G of .
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81 (1) A circle with center 1/2 - i and radius 3 (2) The perpendicular bisector of the segment connecting the points -3i and -1
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Find the equation of the following ellipse.
Ellipse rac{x^{2}}{81}+rac{y^{2}}{9}=1
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Find the area of triangle in each of the following cases.
(1) |\overrightarrow{OA}|=\sqrt{2},|\overrightarrow{OB}}|=\sqrt{3}, \overrightarrow{OA} \cdot \overrightarrow{OB}=2
(2) When the vertices are the three points \( O(0,0), A(1,-3), B(2,2) \)
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(2) The plane passing through the points \( \mathrm{A}(1,0,-5), \mathrm{B}(-1,1,2), \mathrm{C}(2,1,-4) \)
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44 Sequentially (1) \( (2,0), 1 \)
(2) \( \left(1, rac{5}{3} \pi
ight), 3 \)
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Let the pole be O. Find the polar equation of the line that passes through the point with polar coordinates \( \left(\sqrt{3}, rac{\pi}{6}
ight) \) and is perpendicular to the line .
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Find the equation of the ellipse formed by compressing the circle by a factor of 1/2 in the y-axis direction. [Hokkaido Institute of Technology]
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Given the points \( \mathrm{A}(2+i), \mathrm{B}(5+2 i), \mathrm{C}(3+3 i) \) as the vertices of , find the measure of ngle \mathrm{BAC} .
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(2) In the coordinate plane, there are three points \( \mathrm{F}(-5,0), \mathrm{F}^{\prime}(5,0), \mathrm{Q}(x, y) \), where . When the inscribed circle of the triangle touches the x-axis at the point \( (3,0) \), the position of point is determined.
Answer choices:
(0) is constant
(1) is constant
(2) is constant
(3) is constant
Point Q is on the portion of the hyperbola with foci at points and vertices at points \( ( \pm \square, 0) \) where . The equation of the hyperbola is:
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In polar coordinates with pole O, find the polar equations of the following circles. (1) A circle with center at the pole and radius 5 (2) A circle with center at the point A, whose polar coordinates are \( (5,0) \), and radius 5
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What shape will the following parametric curves depict?
(1)
(2)
(3) x=rac{\sin heta}{3}, \quad y=rac{\cos heta}{3}
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Find the foci and asymptotes of the following hyperbolas, and sketch their general shapes.
(1) rac{x^{2}}{25}-rac{y^{2}}{4}=1
(2)
(3)
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Find the foci and asymptotes of the following hyperbolas, and sketch their general shapes.
(1) rac{x^{2}}{25}-rac{y^{2}}{4}=1
(2)
(3)
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Find the equation of the ellipse that satisfies the following conditions:
(2) The foci are at \( (0, \sqrt{5}) \) and \( (0, -\sqrt{5}) \), and the sum of the distances from any point on the ellipse to the foci is 6.
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Answer the following questions about the hyperbola rac{x^{2}}{a^{2}}-rac{y^{2}}{b^{2}}=1 (a>0, b>0).
1. Find the coordinates of the foci of the hyperbola.
2. What is the difference in distances from a point on the hyperbola to the two foci?
3. Find the asymptotes of the hyperbola.
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Find the coordinates of the intersection points of the hyperbola and the line .
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Find the polar coordinates of the point S: (−2, 0)