Can we Show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I expect the imagery on the net can be improved using Mathematica 3D plots.

The double nappe/ sheet of a cone is cut by a plane inclined at angle $ \beta < 2 \alpha $ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs.

The Dandelin spheres are placed in each cone tangentially, cutting plane contacts spheres at respective foci so that their difference is constant.

I can also do this using ContourPlot3D, however you can make it with a great image quality.

Proper choice of $\alpha, \beta$ so that the difference of line segments can be convincingly visualized.