Finding equation of parabola when focus and equations of two perpendicular tangents from any two points on the parabola are given

Question

If the focus of a parabola and the equations of two perpendicular tangents at any two points $P$ and $Q$ on the parabola are given, can we find the equation of the given parabola?

If not, what information can we get from the parabola? (Like length or equation of the Latus rectum, etc)

If the above can be done, is there somewhat of a generalisation when the two tangents are inclined at an angle $\theta$ to each other?

Any hint or a solution would be much appreciated.

Answer 1

The reflection of a parabola's focus in any tangent line gives a point on the directrix. (Why?) Therefore, if you have any two tangents (regardless of the angle they make with one another), then you get two reflected foci, which in turn determine the directrix. With a focus and a directrix, you have a unique parabola. $\square$

Answer 2

The two perpendicular tangents must intersect on the directrix. So if you know the directrix and you know the focus you can find the equation of the parabola. You would probably need to know the axis of symmetry.

Answer 3

The useful property is:

The tangents at ends of any parabola focal ray intersect perpendicularly on its directrix.

From this find out the nice relation between the three slopes

$$ t_1, t , t_2. $$