If one of the slopes of the pair of lines \(\mathrm{ax}^{2}+2 \mathrm{hxy}+\mathrm{by}^{2}=0\) is \(\mathrm{n}\) times the other, then

Let \(m\) be the slope of the lines \(a x^{2}+2 h x y+b y^{2}=0\), then according to question, other slope will be nm.
\(\therefore \quad \mathrm{m}+\mathrm{nm}=\frac{-2 \mathrm{~h}}{\mathrm{~b}}\)
\(\Rightarrow \quad \mathrm{nm}^{2}=\frac{\mathrm{a}}{\mathrm{b}}\)
\(\Rightarrow \quad \mathrm{m}=\pm \sqrt{\frac{\mathrm{a}}{\mathrm{bn}}}\)
\(\therefore\) From Eq. (i), we get
\[
\pm \sqrt{\frac{a}{b n}}(1+n)=\frac{-2 h}{b}
\]
On squaring both sides, we get
\[
\begin{aligned}
\frac{\mathrm{a}}{\mathrm{bn}}(1+\mathrm{n})^{2} &=\frac{4 \mathrm{~h}^{2}}{\mathrm{~b}^{2}} \\
\Rightarrow \quad 4 \mathrm{~h}^{2} \mathrm{n} &=\mathrm{ab}(1+\mathrm{n})^{2}
\end{aligned}
\]