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I found this proof of the reflection property of a hyperbola which is short and uses no algebra (https://www.geogebra.org/m/m6cz5fqR). However, the author says that since the line only intersects the conic section once, it is thus a tangent, using this conclusion to prove the reflective property.

Is his conclusion valid or does it rely on some unsaid assumption? Does the line being angle bisector imply that it has to be the tangent line? If so, would someone kindly state this in a more rigorous manner?

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    $\begingroup$ How about the conic $y=x^2$ and the line $x=1$? $\endgroup$ – Lord Shark the Unknown Sep 1 '18 at 12:23
  • $\begingroup$ ah, good point. what about the case of the hyperbola/ellipse? $\endgroup$ – Conrad Soon Sep 1 '18 at 12:24
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    $\begingroup$ How about the hyperbola $xy=1$ and the line $x=1$? $\endgroup$ – Lord Shark the Unknown Sep 1 '18 at 12:24
  • $\begingroup$ right. i guess i should probably think about questions more before i ask them. so does this mean the authors conclusion is flawed? $\endgroup$ – Conrad Soon Sep 1 '18 at 12:26
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    $\begingroup$ A line intersecting a conic in the projective plane in only one point is a tangent. My examples intersect the conics at points at infinity too. $\endgroup$ – Lord Shark the Unknown Sep 1 '18 at 12:31
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I think you can give a reasonable geometric definition of tangent line to a hyperbola (or ellipse, or parabola) as follows:

A line is tangent to a hyperbola if it intersects the hyperbola at some point $P$ and all the other points of the line are exterior to the hyperbola.

Of course you also need to define interior/exterior points for a hyperbola, but that is fairly obvious and I leave it to you. With the above definition, the proof given at the page you linked should work.

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