Putnam 1976 B4

B-4. For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P$. Prove that $\left(P F_{1}\right)\left(P F_{2}\right) d^{2}$ is constant as $P$ varies on the ellipse, where $P F_{1}$ and $P F_{2}$ are the distances from $P$ to the foci $F_{1}$ and $F_{2}$ of the ellipse.