Question

The maximum area of a rectangle which can be inscribed in an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

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

Answer 1

Answer: (C)

$2ab$

Answer 2

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Area of rectangle $ABCD = (2a \cos \theta)(2b \sin \theta)$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =2 ab \, \sin 2\theta$
This is max. when $\sin 2\theta = 1$
$\therefore$ Max. area = $2ab$.