Question

Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ the area of the triangle formed by the asymptotes and the tangent drawn to it at $(a,0)$ is

• A. $\frac{1}{2}ab$
• B. $ab$
• C. $2ab$
• D. $4ab$

Answer 1

Answer: (B)

Equation of tangent at $(a,0)$ is $x=a$ and the point of intersection of $x=a$ and the asymptotes will be obtained by solving $x=a$ and the equation of asymptotes $bx\pm ay=0$
$\therefore ab\pm ay=0$
$\Rightarrow y=\pm b$
Now, point of intersections are $(a,b)$ and $(a,-b)$
$\therefore$ Area of triangle $=\frac{1}{2}(ab\times 2)=ab$ sq unit