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
