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Geometry and Measurement
Vector Analysis Geometry of Curves and Surfaces - Geometric Transformations (Translation, Rotation, Scaling)
Q.01
"Given the point P'(4,-3), rotate it counterclockwise by -π/3 around the origin to obtain the point Q'(x', y'), find the coordinates of Q'."
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'Find the coordinates of point Q after rotating point P(3, -1) around point A(-1, 2) as the center by an angle of -π/3.'
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'Convert polar coordinates (r, θ) to Cartesian coordinates (x, y).'
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'Find the equation of the curve obtained by translating the curve F(x, y) = 0 parallel to the x-axis by p and parallel to the y-axis by q.'
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'In triangle ABC, with A(a), B(b), C(c), find the vector equations for the following lines: (1) Line passing through the midpoints of sides AB and AC (2) Perpendicular bisector of side BC'
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'(1) Find the coordinates of point B when point A(2,1) is rotated around the origin O by π/4.\n(2) Given that the coordinates of the point obtained by rotating point A(2,1) around point P by π/4 are (1-√2, -2+2√2). Find the coordinates of point P.'
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'For the point P (6, -4, 2), find the coordinates of the following points.'
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'Translation of the given text into multiple languages.'
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'Find the coordinates of point Q when point P(3,1) is rotated around the origin O by π/4.'
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'The parabola y = 1/2 x^2, when shifted parallel to the □ in the x-axis direction and □ in the y-axis direction, touches both the lines y = -x and y = 3x.'
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'Find the coordinates of point Q obtained by rotating point P(-2,3) around the origin by \ \\frac{5}{6} \\pi \.'
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'Find the coordinates of point Q after rotating point P(-2,3) around the origin by 5/6 π.'
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'(1) Translated 2 units along the x-axis and -7 units along the y-axis\n(2) a=3, b=1, c=-5'
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'Parametric representation of an ellipse'
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'What is the equation of the shape formed by moving a curve downwards by -2 units along the y-axis?'
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'Application of vectors to plane geometry'
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"(1) Find the coordinates of point B after rotating point A(2,1) around the origin by π/3. (2) Point Q's coordinates after rotating point A(2,1) around point P by π/3 were Q(3/2-3√3/2,-1/2+√3/2). Find the coordinates of point P."
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'Given a triangle ABC with vertices A(vector a), B(vector b), C(vector c). Find the vector equation of the line passing through the centroid G of triangle ABC and parallel to side AC.'
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'For triangle OAB, if OP=sOA+tOB, find the range of existence of point P when real numbers s and t satisfy the following conditions: (1) s+t=3 (2) 2s+3t=1, s≥0, t≥0 (3) 2s+3t≤6, s≥0, t≥0'
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'Plot the range of movement of point z on the complex plane when the complex number z satisfies |z-1| ≤ |z-4| ≤ 2|z-1|.'
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'Find the coordinates of point B after rotating point A(2,1) around the origin by π/3.'
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'Basic Example 88 Rotate a point around a point that is not the origin'
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'(1) Let point be rotated by an angle around the origin to obtain point . Express in terms of . \n(2) Find the equation of the curve after rotating it around the origin by .'
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'Given the point P(3,-2,1), find the coordinates of the following points:\n(1) The foot A of the perpendicular from point P to the xy plane\n(2) The point B which is symmetrical to point P with respect to the xy plane\n(3) The point C which is symmetrical to point P with respect to the z-axis\n(4) The point D which is symmetrical to point P with respect to the origin'
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'In tetrahedron ABCD, let E be the centroid of triangle BCD, and let G be the point that divides segment AE in the ratio 3:1. Prove that the equation AG+BG+CG+DG=0 holds.'
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Standard 69 | Translation of Parabola (1)
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Development 80 | Symmetric Transformation of Graphs
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What kind of movement does the point z-lpha represent?
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Describe the operation of rotating about a point other than the origin on the complex plane, and represent it using complex numbers. For example, show the case of rotating about the point by an angle of (45 degrees).