Perpendicular tricks!

Take a second degree curve $S\equiv b^2x^2 + a^2y^2 -2b^2hx-2a^2ky- a^2b^2+b^2h^2+a^2k^2=0$ Now a line $L\equiv x\cos \alpha + y\sin \alpha - p-h\cos \alpha-k\sin\alpha=0$

intersects the curve $S=0$ and the points of interception makes right angle at the centre of the curve. If $r$ is the radius of the circle, centred at the centre of $S$ , that the line $L=0$ touches, find $r^2$ when $a=\sqrt5$ and $b= \sqrt3$ .

This question is original and designed to frighten.