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

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

Equation of x- axis is y=0

By solving $a{x}^2+2gx+c=0--\left(i\right)$

(i) intersect on x- axis $\Rightarrow$Roots of (i) are equal

$$\begin{gathered} \Rightarrow {\left(2g\right)}^2-4ac=0 \\ \Rightarrow g^2-ac=0\Rightarrow g^2=ac \\ Now ∆=0\Rightarrow abc+afgh-a{f}^2-b{g}^2-c{h}^2=0 \\ \Rightarrow abc+2fgh-a{f}^2-abc-c{h}^2=0 \\ \Rightarrow 2fgh=a{f}^2+c{h}^2 \end{gathered}$$