A hyperbola has its centre at the origin, passes through the point...

Equation of hyperbola having tranverse axis along $x - axis$ is $\dfrac{{x}^{2}}{{a}^{2}} - \dfrac{{y}^{2}}{{b}^{2}} = 1$

From the given information, equation of hyperbola can be taken as

$\dfrac{{x}^{2}}{4} - \dfrac{{y}^{2}}{{b}^{2}} = 1,$ since $a = 2$



Since it passes through $\left(4, 2\right),$

Hence ${b}^{2} = \dfrac{4}{3}$

Now $e = \sqrt{1 + \dfrac{{b}^{2}}{{a}^{2}} =}\sqrt{1 + \dfrac{1}{3}} = \dfrac{2}{\sqrt{3}}$