Any help on how to solve this ellipsoid-plane-cone intersection problem or just even how to approach it will be greatly appreciated.
All vectors are in $\mathbb{R}^3$ and I am trying to find $\mathbf{r} = [x; y; z ]$; all other vectors are known. The norm is the $\ell_2$ norm. I've simplified it as much as I can and tried Matlab's symbolic solver (mupad I think) but have had no luck. I've also tried first solving for $x$ and $y$ in terms of $z$ and then seeing if I could simplify things for the solver.
$$ \begin{align} \Vert \mathbf{r} - \mathbf{r}_x \Vert + \Vert \mathbf{r} - \mathbf{r}_r \Vert & = a \\ \mathbf{r}^\mathrm{T} \mathbf{p} & = b \\ \mathbf{v}_x^\mathrm{T} \frac {\mathbf{r} - \mathbf{r}_x} {\Vert \mathbf{r} - \mathbf{r}_x \Vert } + \mathbf{v}_r^\mathrm{T} \frac {\mathbf{r} - \mathbf{r}_r} {\Vert \mathbf{r} - \mathbf{r}_r \Vert } & = c \end{align} $$
There are three equations for three unknowns and I know that there are two sets of solutions. I was previously able to get a solutions for the case where $\mathbf{r}_x = \mathbf{r}_r$, but this case has me stumped.