Consider $S=0$ to be an ellipse.
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$
The director circle of this ellipse will be $C=0$. $$x^{2}+y^{2}=a^{2}+b^{2}$$
Consider a point P on $C$ and make two tangents to $S$ touching it at two points A and B.
Let $O$ be the center of the ellipse (it is the origin in this case). Let the feet of the perpendiculars drawn from points $O$ and $P$ to $line AB$ be $M$ and $N$ respectively.
We get the result that $$l(OM).l(PN)=\frac{a^{2}b^{2}}{a^{2}+b^{2}}$$
Which is a constant!
To prove this, I used the cord of contact formula (and the parametric coordinates for $P$) and found the perpendicular distances from $O$ and $P$ to $line AB$ but there was an issue during the multiplication.
Any kind of hint to complete this proof would be appreciated.