$Ifthepairoflines{a}^2\left(x^2+y^2\right)={\left(ax-by\right)}^{2}areperpendiculartoeachother,then$

$\begin{array}{l}GivenEquation{a}^2{x}^2+a^2{y}^2=a^2{x}^2+b^2{y}^2-2abxy\\ \Rightarrow \left(a^2-b^2\right)y^2+2abxy=0\\ Thelinesareperpendiculariff\\ x^{2}co.eff+y^{2}co.eff=0\Rightarrow a^2-b^2=0\\ \Rightarrow a^2=b^2\\ \Rightarrow a=\pm b\end{array}$