If two lines represented by the equation ${\mathrm{ax}}^3+{\mathrm{bx}}^2\mathrm{y}+{\mathrm{cxy}}^2+{\mathrm{dy}}^3=0$ are at right angles then ${\mathrm{a}}^2+{\mathrm{d}}^2+\mathrm{ac}+\mathrm{bd}=$

Put$\mathrm{y}=\mathrm{mx} in {\mathrm{ax}}^3+{\mathrm{bx}}^2\mathrm{y}+{\mathrm{cxy}}^2+{\mathrm{dy}}^3=0$

 $$\begin{gathered} \Rightarrow {\mathrm{dm}}^3+{\mathrm{cm}}^2+\mathrm{bm}+\mathrm{a}=0 \\ \therefore {\mathrm{m}}_1{\mathrm{m}}_2{\mathrm{m}}_3=-\frac{\mathrm{a}}{\mathrm{d}} \end{gathered}$$ 

Two lines are $\perp \text{ re then }{\mathrm{m}}_1{\mathrm{m}}_2=-1$

${\mathrm{m}}_3=\frac{\mathrm{a}}{\mathrm{d}}\text{ put }\mathrm{m}=\frac{\mathrm{a}}{\mathrm{d}}$ in above equation.

$$\begin{gathered} d{\left(\frac{a}{d}\right)}^3+c{\left(\frac{a}{d}\right)}^2+b\left(\frac{a}{d}\right)+a=0 \\ \Rightarrow a^3+c{a}^2+abd+a{d}^2=0 \\ \Rightarrow {\mathrm{a}}^2+{\mathrm{d}}^2+\mathrm{ac}+\mathrm{bd}=0 \end{gathered}$$