2021 AMC 10A Problem 24

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}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.