For 0 < θ < π/2, if the eccentricity of the hyperbola x^2 − y^2cosec^2θ = 5 is

Question

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) \(\frac{\pi}{4}\)

Answer 1

Correct option is (3) \(\frac{\pi}{3}\)

\(e_{h}=\sqrt{1+\sin ^{2} \theta}\)

\(e_{c}=\sqrt{1-\sin ^{2} \theta}\)

\(e_{h}=\sqrt{7} e_{c}\)

\(1+\sin ^{2} \theta=7\left(1-\sin ^{2} \theta\right)\)

\(\sin ^{2} \theta=\frac{6}{8}=\frac{3}{4}\)

\(\sin \theta=\frac{\sqrt{3}}{2}\)

\(\theta=\frac{\pi}{3}\)