AMC 12 2013a Problem 8

Given that $x$ and $y$ are distinct nonzero real numbers such that $x+\tfrac{2}{x} = y + \tfrac{2}{y}$, what is $xy$?

\textbf{(A)}\ \frac{1}{4} \textbf{(B)}\ \frac{1}{2} \textbf{(C)}\ 1 \textbf{(D)}\ 2 \textbf{(E)}\ 4