Suppose I am given with point $P (x, y)$ and the equation of the circle, $C\equiv x^2 + y^2 + 2gx + 2fy + c = 0$. The centre of this circle is $C (-g, -f)$ and its radius, $r = \sqrt{g^2 + f^2 - c}$. Given these data can points $A$ and $B$ be determined?
PAC is a right triangle, so $PA^2 + AC^2 = PC^2$. Since you know AC and PC, this determines PA.
A is at the intersection of the circles center C radius CA and center P radius PA.
You should be able to do the rest.