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)
Best Answer
PERHAPS THIS HELPS
I've been thinking about your problem for a while. I wouldn't say that I could solve it. However, I have an idea that I would like to share with you.
You certainly remember the Dandelin spheres which may have something to do with your conjecture. The following figure depicts two cones (a black one and a grey one). These cones share the vertical axis and a Dandelin sphere. Tangent to this sphere there is a blue plane whose edge can be seen in the figure:
The blue plane and the black cone intersect in a parabola. Also, the blue plane intersects with the other cone in a hyperbola. The parabola's focus and one of the foci of the hyperbola are common.
My conjecture is that this is how to imagine your couple of parabola and hyperbola with a common focus.
Then, I don't know how to proceed.