NL-NWO 1993 Problem 4

Let $C$ be a circle with center $M$ in a plane $V$ , and $P$ be a point not on the circle $C$ . $(a)$ If $P$ is fixed, prove that $AP^2+BP^2$ is a constant for every diameter $AB$ of the circle $C$ . $(b)$ Let $AB$ be a fixed diameter of $C$ and $P$ a point on a fixed sphere $S$ not intersecting $V$ . Determine the points $P$ on $S$ that minimize $AP^2+BP^2$ .