Converse of power of a point for conic sections.

Question

Let $ ax^2+2hxy+by^2=1$ be the equation of a conic section and $ P $ be a point not on the curve. Any line through $ P $ cuts the conic section at $ Q $ and $ R $ such that $ PQ.PR $ is constant. Show that the conic section is a circle.

It's essentially a converse of a power of a point theorem, given that the shape is a conic section. I tried to approach the problem by writing the coordinates of $ Q$ and $ R $ in terms of coordinates of $ P $. Then trying to show that $ h=0$ and $ a=b$. But it seems a cumbersome approach. Any other idea?

Answer 1

The converse for power of a point is true for any $4$ points on the plane. Let $A$, $B$, $C$, $D$ be four distinct points. Let lines $AB$ and $CD$ intersect at $P$. If $P$ lies on the both line segments $AB$ and $CD$ or $P$ lies on neither segments then $PA \cdot PB=PC \cdot PD$ implies that $A$, $B$, $C$, $D$ is concyclic.

You can prove this result by using similar triangles and the definition of cyclic quadrilateral.