Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Equation of the tangent at $\theta$ is $\frac{xsec\theta }{a}-\frac{ytan\theta }{b}=1\to \left(1\right)$
Equation of the asymptotes are $bx-ay=0\to \left(2\right)$
$bx+ay=0\to \left(3\right)$
vertices of the triangle formed by these three lines are $O=(0,0),$
A$(\frac{a}{sec\theta -tan\theta },\frac{b}{sec\theta -tan\theta })$ and B=$(\frac{a}{sec\theta +tan\theta },\frac{-b}{sec\theta +tan\theta })$
$\therefore$ Area of the triangle OAB is ab