(1) Let the center of the circle be C, the coordinates (x, y) with radii R and C, let the midpoint of MN be D, and connect CD and CM, then CD = Y. In △CDM, CD⊥MN, CM 2 = DM 2+CD 2 is R 2 = A 2 and then △ABC. Ac 2 = BC 2+AB 2, that is, R 2 = X 2+(A-Y) 2, then through the above two equations, the center trajectory equation is: X 2 = 2AY.

(2) because ∠ man = 45, ∠ mcn = 90 (the nature of a circle: the central angle of the same chord passing through the circle = twice the circumferential angle), and because MN=2a, if the midpoint d of MN is taken, then CD⊥MN is at the right angle △CDM, cm = MD = A.