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Question

The point (α,β) on the ellipse 4x2+3y2=12, in the first quadrant, so that the area enclosed by the lines y=x, y=β,x=α and the xaxis is maximum, is

A
None of the above
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B
(3,0)
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C
(32,1)
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D
(12,113)
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Solution

The correct option is C (32,1)
Equation of the ellipse is x23+y24=1
Let point P be(3cosθ,2sinθ),θ (0,π2)
Clearly, line PQ isy=2sinθ,
line PR isx=3cosθand OQ is y=x, and Q is (2sinθ,2sinθ)
Z= area of the region PQORP (trapezium)
=12(OR+PQ)PR
=12(3cosθ+(3cosθ2sinθ))2sinθ
=(23cosθsinθ2sin2θ)
=(3sin2θ+cos2θ1)
=2cos(2θπ3)1
Which is maximum when cos(2θπ3)=1 or
2θπ3=0θ=π6
Hence, point P is (32,1)

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