Let's say we are given a general conic's eqn: $ax^2+by^2+2gx+2fy+2hxy+c=0$, and also a fixed point $P(r,s)$. Now I draw a pair of tangents to the conic from $P$. Let these tangents meet the conic at points $Q$ and $R$. Now normals to the conic are drawn at $Q$ and $R$. Let these normals intersect at $M$. Find the exact coordinates of $M$ in terms of $a,b,g,f,h,c,r,s$.
Clearly, these coordinates shall be unique. But despite trying numerous approaches, I haven't succeeded in finding these coordinates. One approach was to find the eqn. of pair of tangents $POT$ by using $SS_1=T^2$ and the eqn. of common chord $QR$. But then finding the intersection points of $POT$ and $QR$ to get the coordinates of $Q$ and $R$ becomes very clumsy, let alone the aftermath of writing eqns. of normals at those points and finally finding their intersection. So I just wanted to know if at all there is an easier method that I might be missing...