Question:
<p>For <span class="math-tex">\(0 \lt \theta \lt \pi / 2\)</span>, if the eccentricity of the hyperbola <span class="math-tex">\(x^{2}-y^{2} \operatorname{cosec}^{2} \theta=5\)</span> is <span class="math-tex">\(\sqrt{7}\)</span> times eccentricity of the ellipse <span class="math-tex">\(x^{2} \operatorname{cosec}^{2} \theta +y^{2}=5\)</span>, then the value of <span class="math-tex">\(\theta\)</span> is:</p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{6}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{5 \pi}{12}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{4}\)</span></p>
Step-by-Step Solution
Key Concept: Convert the given equations into standard forms to express the eccentricities of both the hyperbola and ellipse as functions of sin θ.
<p><span class="math-tex">$e_{h}=\sqrt{1+\sin ^{2} \theta}, e_{c}=\sqrt{1-\sin ^{2} \theta}$</span><br />
Given, <span class="math-tex">$e_{h}=\sqrt{7} e_{c}$</span><br />
<span class="math-tex">$1+\sin ^{2} \theta=7\left(1-\sin ^{2} \theta\right)$</span><br />
<span class="math-tex">$\Rightarrow \sin ^{2} \theta=\frac{6}{8}=\frac{3}{4}$</span><br />
<span class="math-tex">$\Rightarrow \sin \theta=\frac{\sqrt{3}}{2}$</span><br />
Hence, <span class="math-tex">$\theta=\frac{\pi}{3}$</span>.</p>
Correct Answer: C