Given, equation of hyperbola
$x^{2}-y^{2}=a^{2}$...(i)
If $P\left(x_{1}, y_{1}\right)$ is a point on Eq. (i), then
$x_{1}^{2}-y_{1}^{2}=a^{2}$...(ii)
Now, $S P=e x_{1}-a$
and $S P'=e x_{1}+ a$
$\therefore S P \cdot S P'=\left(e x_{1}-a\right)\left(e x_{1}+a\right)$
$=e^{2} \cdot x_{1}^{2}-a^{2}$
$=2 x_{1}^{2}-a^{2}\left(\text { since, for } x^{2}-y^{2}=a^{2}, e=\sqrt{2}\right)$
$=2 x_{1}^{2}-\left(x_{1}^{2}-y_{1}^{2}\right)$[using Eq. (i)]
$=x_{1}^{2}+y_{1}^{2}$