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- Jan 30, 2012

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Does anyone have a reference to a proof of the reflective property of a hyperbola? I need a proof that uses the geometric definition of a hyperbola as the locus of points $X$ such that $|XF_1-XF_2|=2a$ for some fixed points $F_1$ and $F_2$ and a positive constant $a$. The proof may also use elementary geometry but, preferably, no heavy algebra. The reflective property of a hyperbola says that a ray issued from one of the foci and reflected from the hyperbola is seen as issued from the other focus. I have a corresponding proof for an ellipse, but I looked through two of my textbooks and the first page of Google results and did not find a suitable proof for a hyperbola.

Thank you.