Conic Sections · Superchallenge problem

The ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ has foci $F_1 = (-3, 0)$ and $F_2 = (3, 0)$. Let $\ell$ be the line tangent to the ellipse at the point $\left(4, \dfrac{12}{5}\right)$. Find the product of the distances from $F_1$ and from $F_2$ to $\ell$.