Suppose you have an ellipsoid given by the set,
$$\left\{ x \in\mathbb{R}^3 \mid x^TQx = 1 \right\}$$
where $Q = \mbox{diag}(a,b,c)$. Is there a way to parametrize the set
$$\left\{ x \in \mathbb{R}^3 \mid \|p-x\| = k, x^TQx = 1 \right\}$$
where $Q = \mbox{diag}(a,b,c)$, $p\in\mathbb{R}^3$, and $k$ is a real number. In the case I care about, $a=b$, but I am unsure if that simplifies the problem much, and the general case might be helpful to others. Notice if $a=b=c$ the set is a circle and it is relatively easy to parametrize.