Average Level Geometry Problem

The project is about a relatively simple geometry problem. What I require is a *detailed* proof that: For every circle at (X0 , 0) and with X0 always greater than r, there is a circle at (0,Y0(φ)) with a radius R(φ) that intersects the first circle *perpendicularly* at point (x',y'), for any angle φ, and always lands perpendicularly on the Y axis. Note that the first circle always has its center on the X axis and the second circle always has its center on the Y axis. Feel free to contact me for any clarifications before bidding.