If (l ) and (b ) are respectively the length and breadth of the rectangle of greatest area that can be inscribed in the ellipse (x^2+4 y^2=64 ), then ((l, b)= )

Step-by-step solution

Let (P Q R S ) be a rectangle inscribed in the ellipse ( x^2 64 + y^2 16 =1 ). Let the coordinate of (P ) be ((8 , 4 ) ). Then, the coordinates of (Q, R ) and (S ) are ((-8 , 4 ),(-8 , 4 ) ) and ((8 ,-4 ) ) respectively. Let (A ) be the area of rectangle (P Q R S ). Then ( gathered A=P Q R S A=16 8 =64 20 d A d =128 2 and d^2 A d ^2 =-256 2 gathered ) The critical numbers of (A ) are given by ( d A d =0 ) ( d A d =0 4 a b 2 =0 ) ( 2 =0 2 = 2 ) or ( 3 2 ) ( = 4 ) or ( 3 4 ) Clearly, ( ( d^2 A d ^2 )_ = 4 =-256 2 =-2