ILL 1988 Problem 26

The circle $x^2+ y^2 = r^2$ meets the coordinate axis at $A = (r,0), B = (-r,0), C = (0,r)$ and $D = (0,-r).$ Let $P = (u,v)$ and $Q = (-u,v)$ be two points on the circumference of the circle. Let $N$ be the point of intersection of $PQ$ and the $y$ -axis, and $M$ be the foot of the perpendicular drawn from $P$ to the $x$ -axis. If $r^2$ is odd, $u = p^m > q^n = v,$ where $p$ and $q$ are prime numbers and $m$ and $n$ are natural numbers, show that

$$|AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8.$$