Question

Let the eccentricity of the hyperbola with the principal axes along the coordinate axes and passing through $\left(3,0\right)$ and $\left(3 \sqrt{2} , 2\right)$ is $e$ , then the value of $\left(\frac{e^{2} + 1}{e^{2} - 1}\right)$ is equal to

Answer 1

One vertex of the hyperbola is $\left(3,0\right).$
So, assuming the equation of hyperbola as $\frac{x^{2}}{9}-\frac{y^{2}}{b^{2}}=1$
and passing through $\left(3 \sqrt{2} , 2\right)$ , we get,
$\Rightarrow \frac{9 \left(2\right)}{9}-\frac{4}{b^{2}}=1\Rightarrow 1=\frac{4}{b^{2}}\Rightarrow b^{2}=4$
$\Rightarrow e^{2}=1+\frac{b^{2}}{a^{2}}=1+\frac{4}{9}=\frac{13}{9}$
$\Rightarrow \frac{e^{2} + 1}{e^{2} - 1}=\frac{\frac{13}{9} + 1}{\frac{13}{9} - 1}=\frac{22}{4}=\frac{11}{2}=5.5$