CMO 1994 Problem 4

Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $AB$ . Suppose that the line through $PA$ cuts $\Omega$ again at $U$ , and the line through $PB$ cuts $\Omega$ at $V$ . Note that in case of tangency, $U$ may coincide with $A$ or $V$ might coincide with $B$ . Also, if $P$ is on $\Omega$ then $P=U=V$ . Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some $0\le s,t\in \mathbb{R}$ . Determine $\cos \angle APB$ in terms of $s,t$ .