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)
B
(0,2)
C
(5,23)
D
(10,23)

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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