This is a variation of Alhazen's Billiard Problem.
Suppose we have a semicircular billiards table of radius r centered at the origin O, and a billiard ball placed somewhere on the 'x-axis' of the table. Let us call this point P, with coordinate $(0,p)$, with the stipulation that $0$ < P < r. Let the distance OP be $p$. On what point on the table should we aim at such that the billiard ball will bounce off the edge of the table once and into the other 'end' of the x-axis at $(0,-r)$?
Is it also possible to generalize this for any point $(x,y)$ in the circle?