 Intersections of concentric hyperbolas

For the two centred hyperbolas $x^2-y^2=c$ and $4x^2-y^2=d$, with nonzero real $c,d$, eliminating their quadratic terms givesThus they meet in four real points if both quantities are positive, two if one is zero and the other positive, and none if either is negative. Both quantities cannot vanish when $c,d\ne 0$. The choices $(c,d)=(1,5),(1,4),(1,1)$ respectively realize these three counts. This reduction to a linear system in $x^2,y^2$ is useful for intersections of axis-aligned conic sections; the independent sign choices must be counted after solving that linear system.