Conditional distribution of $\angle X$ for $X \in \mathbb{R}^2 \sim \mathcal{N}(\mu, \Sigma)$ given that $|X| = a$

Question

I have a random variable $X \in \mathbb{R}^2$ with an overall distribution $X \sim \mathcal{N}(\mu, \Sigma)$. I would like to be able to specify the conditional distribution of $\angle X$ given that $|X| = a$. How would one do this?

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My thought is that $[\angle X \ | \ |X| = a]$ might be specified as a "wrapped normal" distribution. But I have no idea how to specify the parameters.