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While going through a reference book, I happened to stumble on this question.

If PSQ and PS'R are the focal chords of a hyperbola having foci S and S' such that $|\frac{\text{PS}}{\text{SQ}}-\frac{\text{PS'}}{\text{S'R}}| = 4$, then show that the orthocenter of $\Delta$PQR lies on the hyperbola.

I attempted this question by parameterizing P but I had to deal with half angles which then became lengthy and annoying, further I was unable to find the coordinates of Q and R. I would like a more elegant solution than parameterizing because it is a very very bad idea to bash this problem...

Puneet Singh
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1 Answers1

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You can check that $\left|\frac{\text{PS}}{\text{SQ}}-\frac{\text{PS'}}{\text{S'R}}\right| = 4$ if the hyperbola has eccentricity $e=\sqrt3$, as in figure below.

But triangle $PQR$ doesn't have, in general, its orthocenter $H$ on the hyperbola.

enter image description here

Intelligenti pauca
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