Equation  ${\mathrm{ax}}^3-9{\mathrm{yx}}^2-{\mathrm{y}}^2\mathrm{x}+4{\mathrm{y}}^3=0$ represent 3 straight lines. If 2 of the lines are perpendicular then a =

sine 2 lines are $\perp \mathrm{ler}$ they are ${\mathrm{x}}^2+\mathrm{pxy}-{\mathrm{y}}^2=0$
$\Rightarrow$given equation is $(\mathrm{ax}-4\mathrm{y})\left({\mathrm{x}}^2+\mathrm{pxy}-{\mathrm{y}}^2\right)=0$
Comparing coefficient
$\begin{array}{l}\mathrm{ap}-4=-9,-\mathrm{a}-4\mathrm{p}=-1\\ \Rightarrow \mathrm{p}=\frac{-5}{\mathrm{a}}=\frac{-\mathrm{a}+1}{4}\\ \Rightarrow -{\mathrm{a}}^2+\mathrm{a}=-20\\ {\mathrm{a}}^2-\mathrm{a}-20=0\\ (\mathrm{a}-5)(\mathrm{a}+4)=0\\ \mathrm{a}=-4\text{ or }5\end{array}$