If the pair of lines $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ intersect on the y-axis, then 2fgh =

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

Intersect on y- axis then put x = 0

$\begin{array}{l}\Rightarrow b{y}^2+2fy+c=0\\ D=0\Rightarrow {\left(2f\right)}^2-4\left(b\right)c=0\\ \Rightarrow 4f^2-4bc=0\Rightarrow f^2=bc\end{array}$

$\begin{array}{l}and\left(1\right)representpairoflinethen\Delta =0\\ \Rightarrow abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0\\ \Rightarrow abc+2fgh-abc-b{g}^2-c{h}^2=0\\ \Rightarrow 2fgh=b{g}^2+c{h}^2\end{array}$