Let $\left(4,2\right)=S_1$ and $\left(2,2\right)=S_2$ and

the eccentricity of the hyperbola is $e$, then

$S_1{S}_2=2ae$ and $\left|P{S}_1-P{S}_2\right|=2a$ (where, $2a$ is the length of the transverse axis)
$\Rightarrow e=\frac{S_1{S}_2}{\left|P{S}_1-P{S}_2\right|}=\frac{2}{2\sqrt{2}-2}$
$\Rightarrow e=\frac{1}{\sqrt{2}-1}=\sqrt{2}+1=\tan \frac{3\pi }{8}$