TR-MO 2008 Problem 5

A circle $\Gamma$ and a line $\ell$ is given in a plane such that $\ell$ doesn't cut $\Gamma$ .Determine the intersection set of the circles has $[AB]$ as diameter for all pairs of $\left\{A,B\right\}$ (lie on $\ell$ ) and satisfy $P,Q,R,S \in \Gamma$ such that $PQ \cap RS=\left\{A\right\}$ and $PS \cap QR=\left\{B\right\}$