Length of the subtangent at \(\left(x_{1}, y_{1}\right)\) on \(x^{n} y^{m}=a^{m+n}, m, n>0\), is

Given, \(x^{n} y^{m}=a^{m+n}, \quad m, n>0\)
Taking logarithm on both sides, we get
\[
\begin{gathered}
\log \left(x^{n} y^{m}\right)=\log a^{m+n} \\
\Rightarrow \quad \log x^{n}+\log y^{m}=(m+n) \log a \\
\Rightarrow n \log x+m \log y=(m+n) \log a
\end{gathered}
\]
On differentiating Eq. (i) w.r.t. ' \(x\) ', we get
\(\frac{\mathrm{n}}{\mathrm{x}}+\frac{\mathrm{m}}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=0\)
\(\Rightarrow \quad \frac{\mathrm{m}}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{n}}{\mathrm{x}}\)
\(\Rightarrow \quad \frac{\mathrm{dy}}{\mathrm{dx}}=-\left(\frac{\mathrm{n}}{\mathrm{m}}\right)\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\)
\(\therefore\) Length of subtangent
\(=\frac{y}{d y / d x}\)
\(=\frac{y}{-\left(\frac{n}{m}\right)\left(\frac{y}{x}\right)}\)
\(=\frac{-m x}{n}\)
\(\therefore\) Length of subtangent at \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=\frac{\mathrm{m}}{\mathrm{n}}\left|\mathrm{x}_{1}\right|\)