Let the eccentricity of the hyperbola with the principal axes along...

One vertex of the hyperbola is $\left(3,0\right) .$

So, assuming the equation of hyperbola as $\dfrac{{x}^{2}}{9} - \dfrac{{y}^{2}}{{b}^{2}} = 1$
and passing through \left(3\sqrt{2}, 2\right)$, we get, <br />$⇒ \dfrac{9\left(2\right)}{9} - \dfrac{4}{{b}^{2}} = 1 ⇒ 1 = \dfrac{4}{{b}^{2}} ⇒ {b}^{2} = 4$<br />$⇒ {e}^{2} = 1 + \dfrac{{b}^{2}}{{a}^{2}} = 1 + \dfrac{4}{9} = \dfrac{13}{9}$<br />$⇒ \dfrac{{e}^{2} + 1}{{e}^{2} - 1} = \dfrac{\dfrac{13}{9} + 1}{\dfrac{13}{9} - 1} = \dfrac{22}{4} = \dfrac{11}{2} = 5 . 5