Area of the triangle formed by any tangent to the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ with its
asymptotes is

$$\begin{gathered} \mathrm{Suppose} \mathrm{the} \mathrm{tangent} \mathrm{is} \mathrm{at} (\mathrm{a}, 0) \mathrm{parallel} \mathrm{to} \mathrm{y} \mathrm{axis} \mathrm{then} \\ \mathrm{O}=(0, 0) \mathrm{B}=(\mathrm{a}, \mathrm{b}), \mathrm{C}=(\mathrm{a}, -\mathrm{b}) \\ \mathrm{and} base \mathrm{BC}=2\mathrm{b}, \mathrm{height}=\mathrm{a} \\ \mathrm{Area} of ∆OBC=\frac{1}{2}\times \mathrm{basex} \mathrm{height}=\frac{1}{2}\left(2\mathrm{b}\right)\left(\mathrm{a}\right)=\mathrm{ab} \end{gathered}$$