ISL 1979 Problem 48

In the plane a circle $C$ of unit radius is given. For any line $l$ , a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$ . Let $P$ be a point at distance $r$ from the center of $C$ . One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$ , where $m$ and $n$ are variable mutually orthogonal lines through $P$ . Determine the values of $r$ for which $M(r) > 2$ .