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More specific conics help

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!
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Reply 2
Original post by appledeuce
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?


Because when draw a line from the focus perpendicular to the directrix part of this line will be the minor axis of ellipse.
The symmetry comes from that the points of ellipse are on
a circle with midpoint of focus and a given radius of r and are e*r distance from the directrix (0<e<1) so are on a line parallel with the directrix. The intersections of this parallel line and the circle give the points of ellipse and if there is two points then they will
symmetrical to the line of minor axis

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