Let $D$ be a $d\times d$ diagonal matrix with positive elements on the diagonal, then $x^\top D x = 1$ defines an ellipsoid in $\mathbb R^d$. Let $p$ be a random point on the ellipsoid in the sense that $p/\|p\|$ is uniformly distributed on $S^{d-1}$, i.e., the direction of $p$ is uniformly random. I am looking for the distribution of $\|p\|$.
Any help will be appreciated.