The correct option is
B 64√53Given ellipse 4x2+9y2=144This can be re-written as x236+y216=1
Comparing it to the general form of ellipse we get a=6,b=4
Now eccentricity e is given by e=√1−b2a2=√1−1636=2√56
Now the rectangle is formed by the ends of the latus rectum.
The ends of latus rectum are given by {(ae,b2a),(−ae,b2a),(−ae,−b2a),(ae,−b2a)}
Putting values of a,b,e we gethe ends of latus rectum to be
{(2√5,83),(−2√5,83),(−2√5,−83),(2√5,−83)}
Now to find the area of rectangle, find distances between the two points where area A=l×b′
⇒l=√(4√5)2=(4√5),b′=√(163)2=163
Thus Area ⇒A=(4√5)×163=64√53