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Eccentricity of Hyperbola

If the centre...

Question

If the centre, vertex and focus of a hyperbola be $(0, 0), (4, 0)$ and $(6, 0)$ respectively, then the equation of the hyperbola is

4 x^{2} - 5 y^{2} = 8

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4 x^{2} - 5 y^{2} = 80

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5 x^{2} - 4 y^{2} = 80

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5 x^{2} - 4 y^{2} = 8

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Solution

The correct option is C $5 x^{2} - 4 y^{2} = 80$
Given $a = 4, a e = 6 \Rightarrow e = \frac{3}{2} \Rightarrow \frac{b^{2}}{a^{2}} = e^{2} - 1 = \frac{9}{4} - 1 = \frac{5}{4} \Rightarrow b^{2} = 20$
Therefore, required hyperbola is, $\frac{x^{2}}{16} - \frac{y^{2}}{20} = 1$
$\Rightarrow 5 x^{2} - 4 y^{2} = 80$
Hence, option 'C' is correct.

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