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  Solved  Paper-2005

A) \[\frac{a}{b}\]   

B)                                        \[\sqrt{ab}\]                     

C) \[ab\]                   

D)        \[2ab\]

Correct Answer: D

Solution :

The parametric coordinates of a point that lies on an ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}\,+\frac{{{y}^{2}}}{{{b}^{2}}}\,=1\] are \[(a\cos \theta ,\,b\sin \theta )\]. Let the coordinates of the vertices of rectangle ABCD be andthen Length of rentann and breadth of rectangle, Area of rectangle.  Area of rectangle,      ...(i) \[\therefore \,\,\frac{dA}{d\theta }=2\times 2\,\,ab\,\cos \,2\theta \,\Rightarrow \,\frac{dA}{d\theta }=0\] For maxima or minima, put Now,   Now,    Area is maximum at Maximum area of rectangle =2 ab sq units [from Eq. (i)] Alternate Solution From Eq. (i), Area of rectangle, and A is maximum when Maximum area of rectanglesq units