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Question

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


A
(5,22)
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B
(0,2)
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C
(5,23)
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D
(10,23)
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Solution

The correct option is B $$(5, 2\sqrt {3})$$
$$\dfrac {x^{2}}{12} + \dfrac {y^{2}}{16} = 1$$
$$e = \sqrt {1 - \dfrac {12}{16}} = \dfrac {1}{2}$$
Foci$$=(0.\pm be)$$ $$(0, 2)$$ and $$(0, -2)$$
So, transverse axis of hyperbola $$= 2b = 4$$
$$\Rightarrow b = 2$$
and  $$a^{2} = 1^{2} (e^{2} - 1)$$
$$\Rightarrow a^{2} = 4\left (\dfrac {9}{4} - 1\right )$$
$$\Rightarrow a^{2} = 5$$
$$\therefore$$ It's equation is $$\dfrac {x^{2}}{5} - \dfrac {y^{2}}{4} =- 1$$

$$(5,2\sqrt3)$$ does not satisfy above equation.

Maths

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