已知圆O:x^2+y^2=9,过定点P(1,2)作互相垂直的两弦,AB,CD,则线段AC的中点M的轨迹方程

设圆上两点为A(a,b),C(c,d),M(x,y):a² + b² = 9 (1)c² + d² = 9 (2)x = (a + c)/2,a + c = 2x (3)y = (b + d)/2,b + d = 2y (4)AP² + CP² = AC²(a - 1)² + (b - 2)² + (c - 1)² + (d - 2)² = (a - c)² + (b - d)²简化得:a + c + 2(b + d) - 5 = ac + bd利用(3)(4):x + 2y - 5 = ac + bd (5)直角三角形斜边上的中线等于斜边的一半:MP = AC/24MP² = AC²4(x - 1)² + 4(y - 2)² = (a - c)² + (b - d)² = a² - 2ac + c² + b² - 2bd + d² = (a² + b²) + (c² + d²) - 2(ac + bd)= 9 + 9 - 2(ac + bd)利用(5)并整理得M的轨迹::(x - 1/2)² + (y - 1)² = 13/4