I am working on an optic simulation program. I reached a stage where I am unable to find an algorithm for the desired points.
Here is the required problem,

I figured out how to find the equation of line AA'. 2f is the center of the arc. I have the values of radius of the arc and coordinates of its center.
Question:
Given the equation of line AA', radius of the circle, and coordinates of center,
Find the points at which AA' (diameter) intersects the circle? I need the second solution also for the convex mirror simulation.