Hyperbola Question 1
Question 1 - 2024 (01 Feb Shift 1)
For $0<\theta<\pi / 2$, if the eccentricity of the hyperbola $x^{2}-y^{2} \operatorname{cosec}^{2} \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^{2} \operatorname{cosec}^{2} \theta+y^{2}=5$, then the value of $\theta$ is :
(1) $\frac{\pi}{6}$
(2) $\frac{5 \pi}{12}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{4}$
Show Answer
Answer (3)
Solution
$e _h=\sqrt{1+\sin ^{2} \theta}$
$e _c=\sqrt{1-\sin ^{2} \theta}$
$e _h=\sqrt{7} e _c$
$1+\sin ^{2} \theta=7\left(1-\sin ^{2} \theta\right)$
$\sin ^{2} \theta=\frac{6}{8}=\frac{3}{4}$
$\sin \theta=\frac{\sqrt{3}}{2}$
$\theta=\frac{\pi}{3}$
