Perpendicular tricks!

Geometry Level 5

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 \(b= \sqrt3\).

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