Does proof of the fact that a line only intersects a conic section once imply that it is either a tangent to the conic section?

Question

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?

Answer 1

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.