It is well known that when two tangents to a parabola are perpendicular to each other, they intersect on the directrix. In other words, the intersection point of the two tangents make a straight line, in this case, the directrix,
However, when the two tangents to a parabola intersect at another angle, it seems that the intersection point of the two tangents always forms a hyperbola, regardless of the angle. (The original problem I encountered asked the trace of the intersection point when the angle was 45degrees)
And surprisingly, the hyperbola has the same focus with the parabola.
This can be proven by brute-force algebra, with the same method used to prove the directrix property.
The question I'd like to ask is : Is there a geometric, or maybe an intuitive proof to why the trace of the intersection point forms a hyperbola, and why the focus of the hyperobla and parabola are the same?
It can't be a coincidence that these two conic sections have the same focus, and I assumed that there would be a simple geometric observation that can be made to prove this property without algebra, as many other properties of the conic sections do, but I just can't seem to find it. My math teachers also seem to be stumped with this problem.
Can anyone give me help? I couldn't find any proof regarding this property on google either.
(Sorry for my bad English; it is my first time writing mathematical topics in English, please correct me if anything is wrong)