Conditional mean on a circle

Let $X, Y \sim \mathcal N(0,1)$ be independent. Compute $\mathbb{E}[X \mid X^2 + Y^2]$ .

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Given the radius $X^2+Y^2=r^2$ , the point is uniform on that circle (rotational symmetry).
$X = r\cos\Theta$ with $\Theta$ uniform, so its mean is $0$ .

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