If the lines represented by a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 intersect on the x -axis, which…

Given,

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

put $y=0$

$\Rightarrow a{x}^2+2gx+c=0$

Now roots will be equal to given axis so, $D=0\Rightarrow g^2=ac ......\left(1\right)$

Now for pair of straight line

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

Since $g^2=ac$

So $2fgh=a{f}^2+c{h}^2 .......\left(2\right)$

Also ${\left(2fgh\right)}^2={\left(a{f}^2+c{h}^2\right)}^2$

$\Rightarrow 4f^2{g}^2{h}^2={\left(a{f}^2+c{h}^2\right)}^2$

$\Rightarrow 4f^{2}ac{h}^2={\left(a{f}^2+c{h}^2\right)}^2$

$\Rightarrow {\left(a{f}^2+c{h}^2\right)}^2-4f^{2}ac{h}^2=0$

$\Rightarrow {\left(a{f}^2-c{h}^2\right)}^2=0$

$\Rightarrow a{f}^2=c{h}^2 .........\left(3\right)$

So from equation $\left(1\right), \left(2\right) \& \left(3\right)$ we can say that $abc=2fgh$ is incorrect.