Question

The minimum area of triangle formed by the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and coordinate axes is

• A. ab sq. units
• B. $\frac{a^2 + b^2}{2}$ sq. units
• C. $\frac{(a+b)^2}{2}$ sq. units
• D. $\frac{a^2+ab+b^2}{3}$ sq. units

Answer 1

Answer: (A)

ab sq. units

Answer 2

The equation of the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at $(a \cos \theta, b \sin \theta)$ is $\frac{x \cos \theta}{a} + \frac{y \sin \theta}{b} = 1$. The x-intercept is $\frac{a}{\cos \theta}$ and the y-intercept is $\frac{b}{\sin \theta}$. The area of the triangle formed by the tangent and the coordinate axes is $A = \frac{1}{2} \times \frac{a}{\cos \theta} \times \frac{b}{\sin \theta} = \frac{ab}{2 \sin \theta \cos \theta} = \frac{ab}{\sin(2\theta)}$. The minimum area occurs when $\sin(2\theta)$ is maximum, which is 1. Therefore, the minimum area is $ab$.