The direction cosines of a line satisfy the relation $\lambda \mathit{(}\mathcal{l}\mathit{+}m\mathit{)}\mathit{=}n$ and $\mathrm{mn}+n\mathcal{l}+\mathcal{l}m=0$. The value of $\mathrm{\lambda }$, for which the two lines are perpendicular to each other

Substitutive $\mathrm{l}=-\mathrm{m}+\frac{\mathrm{n}}{\mathrm{\lambda }}$
$\begin{array}{l}\Rightarrow \mathrm{mn}+\mathrm{n}\left(-\mathrm{m}+\frac{\mathrm{n}}{\mathrm{\lambda }}\right)+\mathrm{m}\left(-\mathrm{m}+\frac{\mathrm{n}}{\mathrm{\lambda }}\right)=0\\ \Rightarrow {\mathrm{n}}^2+\mathrm{mn}-{\mathrm{\lambda m}}^2=0\\ \Rightarrow \frac{{\mathrm{n}}^2}{{\mathrm{m}}^2}+\frac{\mathrm{n}}{\mathrm{m}}-\mathrm{\lambda }=0\end{array}$
$\text{ If }\frac{{\mathrm{n}}_1}{{\mathrm{m}}_1}\mathrm{\&}\frac{{\mathrm{n}}_2}{{\mathrm{m}}_2}\text{ are roots, then }\frac{{\mathrm{n}}_1}{{\mathrm{m}}_1}\cdot \frac{{\mathrm{n}}_2}{{\mathrm{m}}_2}=-\mathrm{\lambda }$
$\therefore \frac{{\mathrm{l}}_1{\mathrm{l}}_2}{2}=\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{-1}=\frac{{\mathrm{n}}_1{\mathrm{n}}_2}{\mathrm{\lambda }}\text{ Since the lines are perpendicular }-1+1+\mathrm{\lambda }=2$