We know that ‌ Area. of rectangle ‌‌=l×b A‌=PQ×QR A‌=(2x)(2y)=4xy. for maximizing rectangle area, ‌
dA
dx
‌=4y+4x⋅‌
dy
dx
⇒‌
dA
dx
‌=4y+4x(‌
−x
4y
) ‌=4y−‌
4x2
4y
. for critical point,
‌
dA
dx
=0⇒4y−‌
x2
y
=0⇒x=±2y. Now putting the value of x in curve are get, 4y2+4y2‌=64 ⇒‌‌y2=8‌‌∴y=±2√2‌‌∴x=±4√2. He, points are P(−4√2,+2√2),Q(2√2,4√2) R(4√2,−2√2)&S(−4√2,−2√2) therefore, sides will be, PQ=2x=2×4√2=8√2 QR=2y=2×2√2=4√2.