Find the equation of the hyperbola whose foci are

Question:

Find the equation of the hyperbola whose foci are $(\pm \sqrt{5} 0)$ and the eccentricity is $\sqrt{\frac{5}{3}}$.

 

Solution:

Given: Foci are $(\pm \sqrt{5}, 0)$, and the eccentricity is $\sqrt{\frac{5}{3}}$

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

The eccentricity, $\mathrm{e}=\sqrt{\frac{5}{3}}$

And also given, foci are $(\pm \sqrt{5}, 0)$

That means, ae $=\sqrt{5}$

$\Rightarrow a=\frac{\sqrt{5}}{e}$

$\Rightarrow a=\frac{\sqrt{5}}{\sqrt{\frac{5}{3}}}\left[\right.$ As $\left.e=\sqrt{\frac{5}{3}}\right]$

$\Rightarrow a=\sqrt{3}$

We know that, $e=\sqrt{1+\frac{b^{2}}{a^{2}}}$

Therefore,

 

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