MathNet 2014 Question 88

Let $\mathcal{P}$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $\mathcal{P}$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $A B=20$ and $C D=14$, compute the area of $A B C D$.

Supplemental exact coordinate figure.