By the power of a point on $P$, we get that $16=P D^{2}=(P F)(P E)=2(P E) \Rightarrow P E=8$. However, since $P E=2 P D$ and $\angle F P D=60^{\circ}$, we notice that $P D E$ is a $30-60-90$ triangle, so $D E=4 \sqrt{3}$ and we have $E D \perp D P$. It follows that $D E$ is a diameter of the circle, since tangents the tangent at $D$ must be perpendicular to the radius containing $D$. Hence, the area of the circle is $\left(\frac{1}{2} D E\right)^{2} \pi=12 \pi$