The points $(r\cos \theta, r \sin \theta)$ and $(s \cos(\theta + \frac{\pi}{2}), s \sin(\theta + \frac{\pi}{2}) ) $ lie on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $O$.

Show that:

$\frac{1}{r^2} + \frac{1}{s^2} = \frac{1}{a^2} - \frac{1}{b^2}$

and hence that $e > \sqrt{2}$.

I attempted to put the coordinates in the hyperbola equation and then combine and simplify, but this path doesn't seem to approach the solution. The exercise this question is from is in regards to geometric methods of solving conics question, but I am also unable to see any way in which geometry may be of help here.

Any help as to how to go about this question would be greatly appreciated.

Thank you.


Substituting the coordinates of both points into the hyperbola equation results in: $$ \frac{\cos^2\theta}{a^2}-\frac{\sin^2\theta}{b^2}=\frac1{r^2},\\ \frac{\sin^2\theta}{a^2}-\frac{\cos^2\theta}{b^2}=\frac1{s^2}. $$ Summing the equations one obtains: $$ \frac{1}{a^2}-\frac{1}{b^2}=\frac1{r^2}+\frac1{s^2}. $$


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