Given a SPD matrix $\mathbf{A}$, for example
$\mathbf{A} = \pmatrix{ 5 & 4 \\ 4 & 5}$
defining an ellipsoid $\epsilon_\mathbf{A}$ with semiaxes length given by $\frac{1}{\lambda_i} \mathbf{v}_i$ i.e. the major semiaxis length is defined by the smallest eigenvalue-eigenvector pair, etc.,
how to find the ellipsoid intersections with coordinate axes, i.e. the points
$\mathbf{x} = (x_0, 0)$
and
$\mathbf{y} = (0, y_0)$?