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...
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.