If the slope of one of the lines represented ${a^3}{x^2} + 2hxy + {b^3}{y^2} = 0$ be the square of the other, then $ab(a+b)$ is equal to:

Solution:-

Given:

$a^{3} x^{2}-2 h x y+b^{3} y^{2}=0, \quad a x^{2}+2 h x y+b y^{2}=0$

$a=a^{3}, b=b^{3} \quad$ and $2 h=-2 h \quad m_{1} \cdot m_{2}=\frac{a}{b}$

$\begin{array}{c} m_{1}=m \\ m_{2}=m^{2} \\ m^{3}=\frac{a^{3}}{b^{3}} \\ m=\frac{a}{b} \\ m+m^{2}=\frac{+2 h}{b^{3}} \\ \frac{a}{b}+\frac{a^{2}}{b^{2}}=\frac{2 h}{b^{3}} \end{array}$

Mulfiplying whole equation by $b^{3}$ we get

$a b^{2}+a^{2} b=2 h$

Hence it can also be written as $a b(a+b)=2 h$

Therefore option (a) $2 h$ is the correct choice.