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. $\frac{1}{2} a b$
• B. $a b$
• C. $2 a b$
• D. $4 a b$

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 $b x \pm a y=0$
$\therefore a b \pm a y=0$
$\Rightarrow y=\pm b$
Now, point of intersections are $(a, b)$ and $(a,-b)$
$\therefore $ Area of triangle $=\frac{1}{2}(a b \times 2)=a b$ sq unit