First slide

Hyperbola in conic sections

Question

$e_{1}, e_{2}$ are respectively the eccentricities of the hyperbola $x^{2} - y^{2} {cosec}^{2} θ = 5 and the ellipse x^{2} {cosec}^{2} θ + y^{2}$ $= 5 . If 0 < θ < π / 2 and e_{1} = \sqrt{7} e_{2}, then θ$ is equal to

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{6}$
C

$\frac{π}{3}$
D

none of these

Solution

$e_{1}^{2} = \frac{5 + 5 \sin^{2} θ}{5} = 1 + \sin^{2} θ$
$e_{2}^{2} = \frac{5 - 5 \sin^{2} θ}{5} = \cos^{2} θ$
$\begin{array}{l} e_{1} = \sqrt{7} e_{2} \Rightarrow 1 + \sin^{2} θ = 7 - 7 \sin^{2} θ \\ \Rightarrow \sin θ = \frac{\sqrt{3}}{2} \Rightarrow θ = \frac{π}{3} \end{array}$

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