The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$? $\textbf{(A)} ~\frac{8a^2}{(a+1)^2}\qquad$ $\textbf{(B)} ~\frac{4a}{a+1}\qquad$ $\textbf{(C)} ~\frac{8a}{a+1}\qquad$ $\textbf{(D)} ~\frac{8a^2}{a^2+1}\qquad$ $\textbf{(E)} ~\frac{8a}{a^2+1}$