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 x−axis 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+(√3cosθ−2sinθ))2sinθ =(2√3cosθsinθ−2sin2θ) =(√3sin2θ+cos2θ−1) =2cos(2θ−π3)−1
Which is maximum when cos(2θ−π3)=1 or 2θ−π3=0⇒θ=π6
Hence, point P is (32,1)