= Intersections of concentric hyperbolas
{title2=$x^2-y^2=c,\quad4x^2-y^2=d$}
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 gives
$$
x^2=\frac{d-c}{3},\qquad y^2=\frac{d-4c}{3}.
$$
Thus 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 $c,d\ne0$. 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>.
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