Can find some help me find THE LOCUS OF POINT OF INTERSECTION OF TWO TANGENTS DRAWN TO CIRCLE X^2 + Y^2= A^2?

Forget about algebra. If the angle is a, and the initial radius is R, then the locus is a circle of radius R/(sin(a/2)). If a=60 �, then sin a/2=0.5 and the radius is 2R or 2A with your notations. Just draw a pic it's all obvious.