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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.

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  • $\begingroup$ Are you saying that we know the points $P$ and $Q$ on the parabola, as well as the tangent lines? Or are you saying that we know the tangent lines, but not the specific points of tangency? $\endgroup$ – Blue Jul 16 '15 at 15:24
  • $\begingroup$ Yes. Only the tangent lines are known. $\endgroup$ – user232216 Jul 16 '15 at 15:48
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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. $\square$

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  • $\begingroup$ The reason could be that the vertex is the midpoint of focus and directrix and since you are taking a reflection, slop of the tangent really doesn't matter. So if you consider a particular case of a tangent with slope equal to - 1/(slope of axis), that is tangent at vertex, you obviously get a point on the directrix. So a point on the directrix should satisfy the above. Is this correct? (Thank you for your hints everyone) $\endgroup$ – user232216 Jul 16 '15 at 18:31
  • $\begingroup$ @user232216: I think you might be onto something. However, it might be easier to consider the reflection property of parabolas: a light ray from the focus ($F$) to the point of tangency ($T$) bounces off of the tangent line to become parallel to the axis (say, through some point $P$). So, the normal line at $T$ bisects $\angle FTP$, which implies that the tangent line bisects $\angle FTQ$ (where $Q$ is the point where $TP$ meets the directrix). Then, since $|FT| = |TQ|$ (by the focus-directrix definition of the parabola), we see that $Q$ is the reflection of $F$ in the tangent line. $\endgroup$ – Blue Jul 16 '15 at 18:46
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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.

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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

$$ t_1, t , t_2. $$

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