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An ellipse reflects an incident ray through one focus to the other as reflected ray and its special case of parabola likewise reflects rays parallel to symmetry axis after bouncing to go through its focus.

But what geometric property excludes the hyperbola to have such reflections? I tried to sketch and derive it, but so far no luck. Thanks for showing the ray trace if at all it exists.

It strikes me as odd if no reflective property exists for a hyperbola as well.

EDIT 1

Hyperbola External refection

enter image description here

Narasimham
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    It has them. Take one focus, reflect a line off the hyperbola, follow it out to infinity and back from infinity on the other side, reflect it off the hyperbola again and it hits the other focus. the two infinite segments are part of a single full line. – Will Jagy Dec 03 '15 at 19:20
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    Hyperbolic mirror are used form many years in Cassegrain telescopes. They reflect rays from any direction in the focus: https://en.wikipedia.org/wiki/Cassegrain_reflector – Emilio Novati Dec 03 '15 at 19:24
  • See also : http://neutron.physics.ucsb.edu/docs/hyperbolilc_mirror_info.pdf for panoramic view from hyperbolic mirriros. – Emilio Novati Dec 03 '15 at 19:26
  • Phew! I had known this, I was fixated on a convergent focus, but that should be divergent, reflecting it $away$ .Thanks for comment. – Narasimham Dec 03 '15 at 19:37
  • http://math.stackexchange.com/questions/1110307/alternative-proof-of-the-reflection-property – Will Jagy Dec 03 '15 at 19:52

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