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

Answer: 5.5

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,
$\Rightarrow \frac{9\left(2\right)}{9}-\frac{4}{b^2}=1\Rightarrow 1=\frac{4}{b^2}\Rightarrow b^2=4$
$\Rightarrow e^2=1+\frac{b^2}{a^2}=1+\frac{4}{9}=\frac{13}{9}$
$\Rightarrow \frac{e^2+1}{e^2-1}=\frac{\frac{13}{9}+1}{\frac{13}{9}-1}=\frac{22}{4}=\frac{11}{2}=5.5$