Show that perfect a perfect reflector is a conic section with Fermat's principle

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Homework Statement


Show, using Fermat's principle, that perfect reflecting surfaces are conic sections.


Homework Equations


Equations for the ellipse, parabola and hyperbola


The Attempt at a Solution


Ok, the ellipse seems easy. All rays coming from one focus reflecting to the other focus travels an equal distance if the mirror is an ellipse, since that's the definition of the ellipse.
I'm having problems with the hyperbola. If I understand the question correctly, I'm supposed to show that the rays traveling from one focus to the other (virtual image) are equal in length, but they clearly arent.
Where is the error in my thinking?
 

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