Question

A hyperbola, having the transverse axis of length \(2 \sin \theta\) is confocal with the ellipse \(3 x^{2}+4 y^{2}=12\). Its equation is
(A) \(x^{2} \sin ^{2} \theta-y^{2} \cos ^{2} \theta=1\)
(B)
\(x^{2} \operatorname{cosec}^{2} \theta-y^{2} \sec ^{2} \theta=1\)
(C) \(\left(x^{2}+y^{2}\right) \sin ^{2} \theta=1+y^{2}\)
(D)
\(x^{2} \operatorname{cosec}^{2} \theta=x^{2}+y^{2}+\sin ^{2} \theta\)

Answer 1

Ans : (B)
Hint: Focus of Ellipse is \((1,0)\)
For Hyperbola
\(a_{1}=\sin \theta\)
and \(a_{1} e_{1}=1 \Rightarrow e_{1}=\operatorname{cosec} \theta\)
$$
\begin{aligned}
\Rightarrow \mathrm{b}_{1}^{2} &=\mathrm{a}_{1}^{2}\left(\mathrm{e}_{1}^{2}-1\right) \\
&=\cos ^{2} \theta
\end{aligned}
$$
Equation of Hyperbola is
$$
\frac{x^{2}}{\sin ^{2} \theta}-\frac{y^{2}}{\cos ^{2} \theta}=1
$$

Answer 2

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