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?