Given relation 

$\left(1+a^2\right)\left(1+b^2\right)=4ab$

$\Rightarrow a^2+b^2-2ab=2ab-1-a^2{b}^2$

$\Rightarrow {\left(a-b\right)}^2=-{\left(1-ab\right)}^2$

$\because a>0, a\ne 1$ and $b$ is a positive real number 

$\therefore {\left(a-b\right)}^2\ne 0\ne -{\left(1-ab\right)}^2$, because ${\left(a-b\right)}^2$

and ${\left(1-ab\right)}^2$ are non-negative real numbers. 

$\therefore$ Set $S$ is an empty set.