OIM 1990 Problem 4

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$ , and $M$ is a point on $\Gamma_{1}$ other than $A$ . $\Gamma_{2}$ is a circle tangent to $\ell$ , and also to $\Gamma_{1}$ at $M$ .

a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies.

b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$ , regardless of the position of $M$ .