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Find the area of the greatest rectang...

Find the area of the greatest rectangle that can be inscribed in an ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`

A

ab

B

2 ab

C

`ab//2`

D

`sqrt(ab)`

Text Solution

Verified by Experts

The correct Answer is:
B
Given equation of ellipse, `x^(2)/a^(2)+y^(2)/b^(2)=1`
Let `A (a cos theta, b sin theta)` be any point on elllipse (1 st quadrant)
Coordinate of `B = [ a cos (pi - theta), be sin (pi - theta)]` (2nd quadrant)
`=(-a cos, theta, b sin theta)`
Coordinate of `C=-[a cos(pi+theta), b sin (pi+theta)]` (3rd quadrant)
Coordinate of `D=[a cos (2pi-theta), b sin (2pi-theta)]`(4th quadrant )

Area of the rectangle ABCD
`=(a cos theta+a cos theta)(b sin theta +b sin theta)`
`=2a cos theta xx 2b sin theta = 2ab sin 2theta `
`=2ab xx 1= 2ab`
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