已知圆 x平方+y平方=25,定点C(3,0).A、B为圆上的两个动点,且满足角ACB=90°,求弦AB的中点M的轨迹方

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  • 设A坐标为(x1,y1),B坐标为(x2,y2),

    则M的坐标为(x1+x2)/2,(y1+y2)/2.

    A,B在圆上,且ACB为直角,因此:(用^2表示平方)

    x1^2 + y1^2 = 25

    x2^2 + y2^2 = 25

    (x1-3)^2 + y1^2 + (x2-3)^2 + y2^2 = (x1-x2)^2 + (y1-y2)^2

    => -6(x1+x2) + 18 = -2x1x2 - 2y1y2

    => 6(x1+x2) - 18 = 2x1x2+2y1y2 (左边加50,右边加x1^2+x2^2+y1^2+y2^2)

    => 6(x1+x2) + 32 = (x1+x2)^2 + (y1+y2)^2

    => (x1+x2 - 3)^2 + (y1+y2)^2 = 41

    => ((x1+x2)/2 - 3/2)^2 + ((y1+y2)/2)^2 = 41/4

    所以M的轨迹方程:

    (x-3/2)^2 + y^2 = 41/4