find the greastest area of the rectangle which can be inscribed in the ellipse; x²/a²+y²/b²=1 with sides parallel to the co-ordinate axes

let the vertices of  rectangle be (asinø,bcosø),(asinø,-bcosø),(-asinø,-bcosø),(-asinø,bcosø).(in clockwise sense)

then area of rectangle =2absin2ø

d(area)/dø=2abcosø;

for(maxima) 2ab cosø=0

so ø=pi/4

also d²a/dø²  <0for(ø=pi/4)

hence area =2ab

let the vertices of  rectangle be (asinø,bcosø),(asinø,-bcosø),(-asinø,-bcosø),(-asinø,bcosø).(in clockwise sense)

then area of rectangle =2absin2ø

d(area)/dø=2abcosø;

for(maxima) 2ab cosø=0

so ø=pi/4

also d²a/dø²  <0for(ø=pi/4)

hence area =2ab