Given the point P (2,2), circle C: x2 + y2-8y = 0, the moving straight line L passing through point P intersects with circle C at two points a and B, the midpoint of line AB is m, and O is the coordinate origin to find the trajectory equation of M

Given the point P (2,2), circle C: x2 + y2-8y = 0, the moving straight line L passing through point P intersects with circle C at two points a and B, the midpoint of line AB is m, and O is the coordinate origin to find the trajectory equation of M

Given that circle C1: x ^ 2 + y ^ 2 + 6x-4 = 0 and circle C2: x ^ 2 + y ^ 2 + 6y-28 = 0 intersect at two points a and B, find the circle C equation with the center of the circle on the straight line X-Y-4 = 0 and passing through two points a and B

Let C equation of circle be x ^ 2 + y ^ 2 + 6x-4 + K (x ^ 2 + y ^ 2 + 6y-28) = 0; (k is not equal to - 1)
Then the coordinates of the center of the circle are: x = - 3 / (1 + k), y = - 3K / (1 + k);
Substituting the line X-Y-4 = 0, we can get 3 (k-1) / (K + 1) - 4 = 0;
The solution is k = - 7;
Substituting into the equation of circle C, we get: x ^ 2 + y ^ 2-x + 7y-32 = 0