Find the points at which the diameter intersects the circumference of the circle.

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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, enter image description here

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.