For $0 < \theta < \frac{\pi}{2}$, if the eccentricity of the hyperbola $x^{2} - y^{2} \cosec^{2} \theta = 5$…
For $0 < \theta < \frac{\pi}{2}$, if the eccentricity of the hyperbola $x^{2} - y^{2} \cosec^{2} \theta = 5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^{2} \cosec^{2} \theta + y^{2} = 5$, then the value of $\theta$ is:
Solution
Given:
The equation of hyperbola is, $x^{2} - y^{2} \csc^{2}\theta = 5$
$\Rightarrow \frac{x^{2}}{5} - \frac{y^{2}}{5\sin^{2}\theta} = 1$
And the equation of ellipse is $x^{2}\csc^{2}\theta + y^{2} = 5$
$\Rightarrow \frac{x^{2}}{5\sin^{2}\theta} + \frac{y^{2}}{5} = 1$
Here, $a < b$ as $\sin^{2}\theta \leq 1$
So, $e_{H} = \sqrt{1 + \sin^{2}\theta}$ and $e_{E} = \sqrt{1 - \sin^{2}\theta}$
Also given, $e_{H} = \sqrt{7}e_{E}$
$\Rightarrow \sqrt{1 + \sin^{2}\theta} = \sqrt{7}\sqrt{1 - \sin^{2}\theta}$
$\Rightarrow 1 + \sin^{2}\theta = 7 - 7\sin^{2}\theta$
$\Rightarrow 8\sin^{2}\theta = 6$
$\Rightarrow \sin\theta = \frac{\sqrt{3}}{2}$
$\Rightarrow \theta = \frac{\pi}{3}$