For the hyperbola $\frac{x^{2}}{\cos^{2}\alpha} - \frac{y^{2}}{\sin^{2}\alpha} = 1$, which of the following remains constant when $\alpha$ varies?

Question

Dr. Abubacker · Accepted Answer

(d)
Here, $a^{2} = \cos^{2}\alpha$ and $b^{2} = \sin^{2}\alpha$
Now,$\ \ e = \sqrt{1 + \frac{b^{2}}{a^{2}}}\ \ \  \Rightarrow \ \ \ e = \sqrt{1 + \frac{\sin^{2}\alpha}{\cos^{2}\alpha}}$
$\Rightarrow \ \ \ \ \ e = \sqrt{1 + 	an^{2}{\alpha\ \ }\ } \Rightarrow e = \sec\alpha$
Coordinates of foci are $( \pm ae,0)ie,\ ( \pm 1,0)$
Hence, abscissae of foci remain constant when $\alpha$ varies.

Q. For the hyperbola $\frac{x^{2}}{\cos^{2}\alpha} - \frac{y^{2}}{\sin^{2}\alpha} = 1$, which of the following remains constant when $\alpha$ varies?

Explanation

Reviewed by Dr. Abubacker

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