Algebra

$\begin{aligned} \frac {x+y}{4\sqrt{xy}} & = 1 \\ \frac {\frac xy + 1}{4 \sqrt{\frac xy}} & = 1 \\ \frac xy + 1 & = 4 \sqrt{\frac xy} \\ \left( \sqrt{\frac xy} \right)^2 - 4 \sqrt{\frac xy} + 1 & = 0 \\ \implies \sqrt{\frac xy} & = \frac {4 \pm \sqrt{16-4}}2 \\ & = 2 \pm \sqrt 3 \\ \implies \frac xy & = \left( 2 \pm \sqrt 3 \right)^2 \\ & = 7 \pm 4\sqrt 3 \end{aligned}$

Therefore, the answer is $\boxed{7+4\sqrt 3}$