Finding the pdf of $(X+Y)^2/(X^2+Y^2)$ where $X$ and $Y$ are independent and normal 
$X$ and $Y$ are iid standard normal random variables.  What is the pdf of $(X+Y)^2/(X^2+Y^2)$?

I am guessing you would transform into polar coordinates and go from there, but I am getting lost.  Do we need two variable transformations here? 

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In order to keep it simple enough, I'll assume normal distributions like
$\ds{\expo{-\xi^{2}/2} \over \root{2\pi}}$. Lets
$\ds{z \equiv {\pars{x + y}^{2} \over x^{2} + y^{2}}}$:

\begin{align}
\color{#66f}{\large\pp\pars{z}}&=\int_{-\infty}^{\infty}{\expo{-x^{2}/2} \over \root{2\pi}}
\int_{-\infty}^{\infty}{\expo{-y^{2}/2} \over \root{2\pi}}\,
\delta\pars{z - {\bracks{x + y}^{2} \over x^{2} + y^{2}}}\,\dd x\,\dd y
\\[5mm]&={1 \over 2\pi}\ \overbrace{\int_{0}^{\infty}\dd r\,r\expo{-r^{2}/2}}
^{\ds{=}\ \dsc{1}}\
\int_{0}^{2\pi}\delta\pars{z - 1 - \sin\pars{2\theta}}\,\dd\theta
\end{align}

It's clear that $\ds{\pp\pars{z} = 0}$ whenever $\ds{z < 0}$ or $\ds{z > 2}$. Hereafter, I'll assume that $\ds{z \in \pars{0,2}}$:

\begin{align}
\left.\color{#66f}{\large\pp\pars{z}}\right\vert_{z\ \in\ \pars{0,2}}&
={1 \over \pi}\int_{0}^{\pi}\delta\pars{z - 1 - \sin\pars{2\theta}}\,\dd\theta
={1 \over \pi}\int_{-\pi/2}^{\pi/2}
\delta\pars{z - 1 + \sin\pars{2\theta}}\,\dd\theta
\\[5mm]&={1 \over \pi}\int_{0}^{\pi/2}\bracks{%
\delta\pars{z - 1 + \sin\pars{2\theta}} + \delta\pars{z - 1 - \sin\pars{2\theta}}}
\,\dd\theta
\\[5mm]&={1 \over \pi}\int_{-\pi/4}^{\pi/4}\bracks{%
\delta\pars{z - 1 + \cos\pars{2\theta}} + \delta\pars{z - 1 - \cos\pars{2\theta}}}
\,\dd\theta
\\[5mm]&={2 \over \pi}\int_{0}^{\pi/4}\bracks{%
\delta\pars{z - 1 + \cos\pars{2\theta}} + \delta\pars{z - 1 - \cos\pars{2\theta}}}
\,\dd\theta
\\[5mm]&={1 \over \pi}\int_{0}^{\pi/2}\bracks{%
\delta\pars{z - 1 + \cos\pars{\theta}} + \delta\pars{z - 1 - \cos\pars{\theta}}}
\,\dd\theta
\\[5mm]&={1 \over \pi}\int_{0}^{\pi/2}
\delta\pars{\verts{z - 1} - \cos\pars{\theta}}\,\dd\theta
\\[5mm]&={1 \over \pi}\int_{0}^{\pi/2}
{\delta\pars{\theta - \arccos\pars{\verts{z - 1}}} \over \verts{\sin\pars{\theta}}}\,\dd\theta
\\[5mm]&={1 \over \pi}\,{1 \over \root{1 - \verts{z - 1}^{2}}}
={1 \over \pi}\,{1 \over \root{z\pars{2 - z}}}\,,\qquad z \in \pars{0,2}
\end{align}

$$
\color{#66f}{\large\pp\pars{z}}
=\color{#66f}{\large\left\{\begin{array}{lcl}
{1 \over \pi}\,{1 \over \root{z\pars{2 - z}}} & \color{#000}{\mbox{if}} & z \in \pars{0,2}
\\
0 && \mbox{otherwise}
\end{array}\right.}
$$

