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第4章:坐标变换
高三数学 | 高级数学
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7.3 Parametric Equations and Locus 参数方程式与轨迹


(1)
Find the equation of the locus of the centers of circles passing through the points of intersection of the circle x^2 + y^2 – 2x – 4y – 11 = 0 and the line y = x + 3.

求经过圆x^2 + y^2 – 2x – 4y – 11 = 0及直线y = x + 3的交点的圆之圆心的轨迹方程式.

Answer :
x + y = 3
(2)
A variable circle touches the y-axis, and also touches a fixed circle x^2 + y^2 – 8x + 2y + 13 = 0 externally. Find the equation of the locus of the centre of the circle.

一动圆与y轴相切, 且与圆x^2 + y^2 – 8x + 2y + 13 = 0外切. 求其圆心的轨迹方程式

Answer :
y^2 – 12x + 2y + 13 = 0
(3)
Find the equation of the locus of the centers of the circles that orthogonal with x^2 + y^2 – 6x + 4y + 5 = 0 and x^2 + y^2 + 2x – 9 = 0.

一圆与二已知圆x^2 + y^2 – 6x + 4y + 5 = 0 及 x^2 + y^2 + 2x – 9 = 0正交, 试求此圆圆心的轨迹的方程式

Answer :
4x – 2y – 7 = 0
(4)
The equation x^2 + y^2 + ax + by + 7 = 0 represents a unit circle. Find the equation of the locus of its centre as a, b vary.

若 x^2 + y^2 + ax + by + 7 = 0 为一单位圆。当a, b 改変时求圆心的轨迹方程式。

Answer :
x^2 + y^2 = 8
(5)
A straight line through the point (0,1) meets the circle C : x^2 + y^2 – 2x – 4y + 4 = 0 at points P_1, P_2 such that the slope of P_1P_2 is k. Find the equation of the locus of the mid-point of P_1P_2, as k varies.

一经过点(0, 1)的直线与圆 x^2 + y^2 – 2x – 4y + 4 = 0 交于点 P_1, P_2 且 P_1P_2 斜率为k。當k改変时, 求 P_1P_2 中点轨迹方程式。

Answer :
x^2 + y^2 – x – 3y + 2 = 0
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