AMC 12 2020b Problem 25

For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that

$$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$$ What is the maximum value of $P(a)?$

\textbf{(A)}\ \frac{7}{12} \textbf{(B)}\ 2 - \sqrt{2} \textbf{(C)}\ \frac{1+\sqrt{2}}{4} \textbf{(D)}\ \frac{\sqrt{5}-1}{2} \textbf{(E)}\ \frac{5}{8}