If $x^2+\alpha y^2+2\beta y=a^2$ represents a pair of perpendicular lines then $\beta =$

Given Equation  $x^2+\alpha y^2+2\beta y=a^2$  ---(i)

Comparing with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

$a=1, h=0, b=\alpha ,g=0, f=\beta , c=-a^2$

(i) represent pair of  $\perp r$ lines

$\Rightarrow 1+\alpha =0\Rightarrow \alpha =-1$

and $$\begin{gathered} ∆=0\Rightarrow abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0 \\ \Rightarrow 1\left(\alpha \right)(-{\alpha }^2)+0-1\left({\beta }^2\right)-\alpha \left(0\right)-0=0 \\ \Rightarrow -\alpha a^2-{\beta }^2=0 \\ \Rightarrow -(-1)a^2-{\beta }^2=0 \\ \Rightarrow a^2={\beta }^2 \\ \Rightarrow \beta =a \end{gathered}$$