I'm looking at conic sections, and I've been wondering why conic sections are symmetrical about their minor axis (for instance, why isn't the ellipse egg-shaped, not symmetrical?). I considered the ellipse first, and using Dandelin spheres, have proved nearly every property of an ellipse - except the minor symmetry of it.
So here's my question: Why is the directrix-focus distance equal on both sides?
An equivalent question uses the directrix-vertex distance, or the vertex-focus distance on both sides - these are all sufficient to prove symmetry, but I can't seem to make it work! Any help would be very much appreciated, thanks!