Conditioning a normal on the radius squared

Take two independent standard normal variables $X$ and $Y$. Given the value of $X^2+Y^2$, what is the conditional expectation of $X$? That is, find $\mathbb{E}\left[X \mid X^2+Y^2\right]$.

$0$. Replacing $X$ by $-X$ leaves the joint law of $(X,Y)$ and the value of $X^2+Y^2$ unchanged, so the conditional mean of $X$ must equal the conditional mean of $-X$. A quantity equal to its own negative is zero.