Question

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

• A. ab/2
• B. ab
• C. 2ab
• D. 4ab

Answer 1

Answer: (B)

ab

Answer 2

The equation of the hyperbola is given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The asymptotes are $y = \pm \frac{b}{a}x$. The equation of the tangent at (a, 0) is $\frac{ax}{a^2} - \frac{0y}{b^2} = 1$, which simplifies to $x = a$.

The intersection points of the tangent $x = a$ with the asymptotes $y = \pm \frac{b}{a}x$ are $(a, b)$ and $(a, -b)$.

The vertices of the triangle are (0, 0), (a, b), and (a, -b). The base of the triangle along the line x = a has length 2b, and the height of the triangle is a.

The area of the triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2b \cdot a = ab$.