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

$\text{ Let }f(x,y)=a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

$$\begin{gathered} \sin ce given lines intersect on y-axis then length of y-intercept=0 \\ \Rightarrow f^2=bc \\ Now ∆=0\Rightarrow abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0 \\ sub f^2=bc then \\ 2fgh=b{g}^2+c{h}^2 \end{gathered}$$