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 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 intercepts on the coordinate axes are $X = \frac{a}{\cos \theta}$ and $Y = \frac{b}{\sin \theta}$. The area of the triangle formed by the tangent and axes is $A = \frac{1}{2}|XY| = \frac{ab}{|\sin(2\theta)|}$. The minimum area is obtained when $|\sin(2\theta)|$ is maximum (i.e., 1), which gives $A_{min} = ab$.