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Q.

A hyperbola having the transverse axis of length $2 \sin θ,$ is confocal (same foci) with ellipse
$3 x^{2} + 4 y^{2} = 12$ . Then its equation is

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a

$x^{2} \sin^{2} θ - y^{2} \cos^{2} θ = 1$

b

$x^{2} \cos e c^{2} θ - y^{2} \sec^{2} θ = 1$

c

$x^{2} \sec^{2} θ - y^{2} \cos e c^{2} θ = 1$

d

$x^{2} \cos^{2} θ - y^{2} \sin^{2} θ = 1$

answer is A.

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Detailed Solution

$\frac{x^{2}}{4} + \frac{4^{2}}{3} = 1, e = \frac{1}{2}, F o c i = (\pm 1, 0)$
Hyperbola with same foci $\Rightarrow 2 a e = 2$
Given $2 a = 2 \sin θ \Rightarrow e = \frac{1}{\sin θ} = \cos c e θ$
$a = \sin θ, b^{2} = a^{2} e^{2} - a^{2} = 1 - \sin^{2} θ = \cos^{2} θ$

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