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

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

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



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

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

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