The subtangent at any point on the curve $x^m y^n=a^{m+n}$ varies as

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We have,

$x^m y^n=a^{m+n}$

$\Rightarrow m \log _e x+n \log _e y=(m+n) \log _e a $

$\Rightarrow \frac{m}{x}+\frac{n}{y} \frac{d y}{d x}=0$                [Differentiating w.r.t. x]

$\Rightarrow \frac{d y}{d x}=-\frac{m y}{n x}$

∴  Length of the subtangent $=\left|\frac{y}{\frac{d y}{d x}}\right|=\left|y \times \frac{n x}{m y}\right|=\frac{n}{m}|x| \propto x$