How to parametrize the intersection of an ellipsoidal surface and a sphere?

Suppose you have an ellipsoid given by the set,

$$left{ x inmathbb{R}^3 mid x^TQx = 1 right}$$

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 $Q = mbox{diag}(a,b,c)$, $pinmathbb{R}^3$, 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.