The length of the subtangent (if exists) at any point $\mathrm{\theta }$ on the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ is

The given curve is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\begin{array}{r}\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0\\ \Rightarrow \frac{dy}{dx}=-\frac{b^{2}x}{a^{2}y}\end{array}$
$\begin{array}{r}=-\frac{b^2\cdot a\cos \theta }{a^2\cdot b\sin \theta }\\ =-\frac{b\cos \theta }{a\sin \theta }\end{array}$
$$\begin{gathered} \therefore \text{ Length of sub-tangent } \\ \begin{array}{l}=\frac{y}{\frac{dy}{dx}}=\frac{b\sin \theta }{-\frac{b\cos \theta }{a\sin \theta }}\\ =-\frac{a{\sin }^2\theta }{\cos \theta }\end{array} \end{gathered}$$
$\begin{array}{r}=\left|\frac{y}{\frac{dy}{dx}}\right|==\left|-\frac{a{\sin }^2\theta }{\cos \theta }\right|\\ \\ =a{\sin }^2\theta |\sec \theta |\end{array}$