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