Ellipse
Optimization
IIT-JEE 2005 Screening
Grade 11
Question:
The minimum area of the triangle formed by the tangent to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the coordinate axes is:
(1) $ab$
(2) $\frac{a^2 + b^2}{2}$
(3) $\frac{(a + b)^2}{2}$
(4) $\frac{a^2 + ab + b^2}{3}$
Step-by-Step Solution
Key Concept: Tangent at $(a \cos \theta, b \sin \theta)$: $x \cos \theta/a + y \sin \theta/b = 1$. Area of triangle with axes $= \frac{1}{2} \cdot \frac{a}{\cos \theta} \cdot \frac{b}{\sin \theta} = \frac{ab}{\sin 2\theta} \geq ab$. Minimum $= ab$ at $\theta = \pi/4$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)