The transverse axis of a hyperbola is of length $2 a$ and a vertex divides the segment of the axis between the centre and the corresponding focus in the ratio $2 : 1$ . The equation of the hyperbola is

4 x^{2} - 5 y^{2} = 4 a^{2}

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4 x^{2} - 5 y^{2} = 5 a^{2}

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5 x^{2} - 4 y^{2} = 4 a^{2}

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5 x^{2} - 4 y^{2} = 5 a^{2}

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Solution

The correct option is C $5 x^{2} - 4 y^{2} = 5 a^{2}$
We have given $\frac{a}{a e - a} = 2 \Rightarrow e = \frac{3}{2}$
Using $e^{2} = 1 + \frac{b^{2}}{a^{2}}$
$\frac{9}{4} = 1 + \frac{b^{2}}{a^{2}} \Rightarrow b^{2} = \frac{5}{4} a^{2}$
Hence required hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{\frac{5}{4} a^{2}} = 1$
$\Rightarrow 5 x^{2} - 4 y^{2} = 5 a^{2}$
Hence option 'D' is correct.

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