The asymptotes of the hyperbola x 2 a 2 - y 2 b 2 = 1 , with any tangent to the hyperbola form a triangle…

Given hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Asymptotes of hyperbola are $\frac{x}{a}+\frac{y}{b}=0...\left(I\right) , \frac{x}{a}-\frac{y}{b}=0...\left(II\right)$

Tangent at $\left(x_1, y_1\right)$ is $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1....\left(III\right)$

We know that area formed by a triangle with asymptotes and any tangent is equal to $ab$.

$\Rightarrow ab = a^2 \tan \alpha$

$\Rightarrow b =a \tan \alpha$

Eccentricity of hyperbola are $e = \sqrt{\frac{a^2+b^2}{a^2}}$

$\Rightarrow e = \sqrt{\frac{a^2+a^2{\tan }^2\alpha }{a^2}}= \sqrt{1+{\tan }^2\alpha } = \sec \alpha .$