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A Hyperbola whose transverse axis is along the major axis of the conic $\frac{x^{2}}{3} + \frac{y^{2}}{4} = 4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\frac{3}{2}$ , then which of the following points does not lie on it ?

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$(5, 2 \sqrt{3})$

$(0, 2)$

$(\sqrt{5}, 2 \sqrt{2})$

$(\sqrt{10}, 2 \sqrt{3})$

answer is B.

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Detailed Solution

$\frac{x^{2}}{12} + \frac{y^{2}}{16} = 1 \Rightarrow e = \frac{1}{2}$

$\begin{array}{l} s = (0, 2) s^{1} = (0, - 2) \\ L e n g t h o f t h e t r a n s v e r s e a x i s \\ = 2 b = 4 \Rightarrow b = 2 (b > a) \\ a^{2} = 4 (\frac{9}{4} - 1) = 5 \\ ∴ Re q . e q i s \frac{x^{2}}{5} - \frac{y^{2}}{4} = - 1 . \end{array}$

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