3
$\begingroup$

The green line below is tangent drawn at point $P$. This construction uses the fact that the angle $PFT$ is $90$ degrees. But it doesn't give any explanation of why it must be $90$. Is there any simple way to see this with or without calculus?

Intuitively, when $P$ are right above $F$, it is clear that the angle is 90 because $PF$ is vertical and $FT$ is horizontal. As $P$ moves to the right, it seems $T$ also moves upkeep the angle at focus $90$. I'm not that sure how to approach proving things like these... Help appreciated.

enter image description here

$\endgroup$
4
  • $\begingroup$ Video on above construction is here . (Watching this video is not needed to answer my question. I think I've included all the necessary details in the question itself) $\endgroup$
    – AgentS
    Jun 28, 2018 at 5:20
  • 1
    $\begingroup$ I would look for similar or even congruent triangles. If $Q$ is the foot of the altitude from the directrix to $P$, then $PQ=PF$ and the tangent line bisects $\angle{FPQ}$. $\endgroup$
    – amd
    Jun 28, 2018 at 5:35
  • $\begingroup$ Ok.. In triangles $PFT$ and $PQT$ we have $\angle PQT=90$, $PQ = PF$ and $PT=PT$. How do we know the tangent bisects $\angle FPQ$ ? @amd $\endgroup$
    – AgentS
    Jun 28, 2018 at 5:49
  • $\begingroup$ The reflective property. $\endgroup$
    – amd
    Jun 28, 2018 at 6:57

1 Answer 1

2
$\begingroup$

Here's an image that can help you visualize the reflective property (incoming horizontal rays are reflected toward the focus) and why the tangent bisects $\angle FPQ$.

Tangent in P bisects angle FPQ

$\endgroup$
1
  • $\begingroup$ Ah, if we assume that the parabola reflects the incoming horizontal rays through the focus, then by angle of incidence = angle of reflection, the tangent indeed bisects $\angle FPQ$. Thank you :) the graph really helped :) I'll google a bit why the horizontal rays have to go through the focus. Thanks again :) $\endgroup$
    – AgentS
    Jun 28, 2018 at 10:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.