已知圆M:x2;+y2;-2mx-2ny+m2;-1=0与圆N:x2+y2+2x+2y-2=0交于A,B两点,且这两点平

这两点平分圆N的圆周, 则AB为圆N的直径x² + y² + 2x + 2ny - 2 = 0(x + 1)² + (y + 1)² = 4圆心为N(-1, -1), 半径r =2x² + y² - 2mx - 2ny + m² - 1 = 0(x - m)² + (y - n)² = n² + 1圆M的圆心为M(m, n), 半径R= √(n² + 1)显然MNA为直角三角形: MA² = NA² + MN²R² = r² + (m + 1)² + (n + 1)²n² + 1 = 4 + (m + 1)² + (n + 1)²(m+1)² + 2n + 4= 0分别用x, y取代m, n,M的轨迹方程为: (x+1)² + 2y + 4= 0(x+1)² + 2y + 4= (x + 1)² + 2(y + 2) = 0此为顶点为(-1, -2)的抛物线, m = -1, n = -2时, 圆M半径最小, 方程:(x + 1)² + (y + 2)² = 5