Question

The eccentricity of the hyperbola passing through the points $(3,0)$ and $(3\sqrt{2},2)$ is _________

Answer 1

Let the equation of hyperbola be of the form:

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Given that hyperbola passes through point $(3,0)$ and $(3\sqrt{2},2)$

$\therefore \frac{3^2}{a^2}-\frac{0^2}{b^2}=1$

$\Rightarrow \frac{9}{a^2}=1$

$\Rightarrow a^2=9\dots \left(1\right)$

And $\frac{(3\sqrt{2}{)}^2}{a^2}-\frac{2^2}{b^2}=1$

$\Rightarrow \frac{18}{9}-\frac{4}{b^2}=1(\therefore a^2=9)$

$\Rightarrow 2-\frac{4}{b^2}=1$

$\Rightarrow \frac{4}{b^2}=1$

$\Rightarrow b^2=4\dots \left(2\right)$

$\therefore \text{Eccentricity},e=\sqrt{1+\frac{b^2}{a^2}}$

$\Rightarrow e=\sqrt{1+\frac{4}{9}}\left[\text{From equation}\right(1\left)\text{and}\right(2\left)\right]$

$\Rightarrow e=\frac{\sqrt{13}}{3}$