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If $\alpha$, $\beta$, $\gamma$, and $\delta$ are the eccentric angles of four concyclic points on a hyperbola, then prove that their sum is an even multiple of $\pi$.

I tried doing this by writing a general second degree equation that will certainly pass through these four points using the equation of the hyperbola as $x^2/a^2 - y^2/b^2 = 1$ and two chords that pass through these two points each, but the condition is a bit too lengthy. Any advice on simplifying it would be greatly appreciated.

Blue
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Puneet Singh
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1 Answers1

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Hint:

WLOG the equation of the circle $$x^2+y^2+2gx+2fy+c=0$$

Now the four points be $P(a\sec t, b\tan t)$ where $t\in\alpha,\beta,\gamma,\delta$

$$a^2\sec^2t+b^2\tan^2t+2g(a\sec t)+2f(b\tan t)+c=0$$

$$\iff a^2+b^2\sin^2t+2ga\cos t+2fb\sin t\cos t+c\cos^2t=0$$

Now use Weierstrass substitution to form a quadratic equation in $\tan\dfrac t2$

Now use Vieta's formulas to find $$\tan\dfrac{\alpha+\beta+\gamma+\delta}2=\dfrac{\sum\tan\dfrac\alpha2-\sum_{\text{cyc}}\tan\dfrac\alpha2\tan\dfrac\beta2\tan\dfrac\gamma2}{\cdots}=0$$

lab bhattacharjee
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  • [+1] By chance, I found your solution to the analogous result where the hyperbola is replaced by an ellipse (https://math.stackexchange.com/q/623242). It means that there must be a more general solution, using maybe homographies and/or pencils of conics... – Jean Marie Mar 12 '20 at 07:42
  • @JeanMarie, Thanks for reminding that solution and linking to this – lab bhattacharjee Mar 12 '20 at 07:46
  • An interesting connected issue/solution : (https://math.stackexchange.com/q/2904889) – Jean Marie Mar 12 '20 at 07:50
  • Yes this is usually the case with these two as they are both 'central' conics... – Puneet Singh Mar 12 '20 at 09:21