question_answer

Area of the greatest rectangle that can be inscribed in the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is                                       [AIEEE 2005]                                         

A)            \[\sqrt{ab}\]

B)            \[\frac{a}{b}\]

C)                 \[2ab\]

D)                 \[ab\]

Correct Answer: C

Solution :

                              Area of rectangle ABCD                    = \[(2a\cos \theta )\]. \[(2b\sin \theta )=2ab\sin 2\theta \]                                 Hence, area of greatest rectangle is equal to \[2ab\], when \[\sin 2\theta =1\].