If the slope of one of the lines represented by $a{x}^2+2hxy+b{y}^2=0$ is the square of the other, then $\frac{a+b}{h}+\frac{8h^2}{ab}$is

The correct option is D

$6$

Explanation of correct option:

Step 1: Find the equation of pair of straight line :

Let the slope be $m$.

Then the other line will have a slope $m^2$

The equation of line will be,

$$\begin{gathered} (y-mx)\left(y-m^{2}x\right)=0 \\ \Rightarrow m^3{x}^2+y^2-mxy-m^{2}xy=0 \\ \Rightarrow m^3{x}^2-xy\left(m^2+m\right)+y^2=0 \end{gathered}$$

Step 2: Find the given expression :

Comparing the equation $m^3{x}^2-xy\left(m^2+m\right)+y^2=0$ this with the given equation $a{x}^2+2hxy+b{y}^2=0$

$$\begin{gathered} a=m^3 \\ b=1 \\ h=\frac{-({\mathrm{m}}^2+\mathrm{m})}{2} \end{gathered}$$

Now,

$\therefore \frac{a+b}{h}+\frac{8h^2}{ab}$$=\left[\frac{-2\left(m^3+1\right)}{\left(m^2+m\right)}\right]+\frac{2{\left(m^2+m\right)}^2}{m^3}$

$=\frac{\left(-2m^5-2m^2+2m^5+2m^2+6m^4+6m^3\right)}{m^3(m+1)}$

$=\frac{\left(6m^4+6m^3\right)}{m^3(m+1)}$

$=\frac{6m^3(m+1)}{m^3(m+1)}$

$=6$

Hence, option (C) is correct.