Confocal conics

Confocal conics share the same pair of foci. For foci $(\pm c,0)$ with $c>0$, a useful family of ellipses and hyperbolas isFor $\lambda >0$ this is an ellipse with squared semiaxes $c^2+\lambda$ and $\lambda$. For $-c^2<\lambda <0$ it is a hyperbola; in both cases the focal distance is $c$. The parameter endpoints are degenerate cases, excluded from this formula.

At an intersection of members with distinct parameters $\lambda ,\mu$, subtracting their equations givesThis is the dot product of their normal directions, so the two conics meet orthogonally. The resulting ellipse-hyperbola grid underlies elliptic coordinates and useful separation of variables for planar differential equations.