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Question

If x and y are real numbers which satisfy the relation x2+9y24x+6y+4=0, then the maximum value of (4x9y)2 is

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Solution

x2+9y24x+6y+4=0
x24x+4+9y2+6y+1=1
(x2)2+(3y+1)2=1
(x2)2+(y+13)219=1
which is an equation of ellipse having centre at (2,13).

If P(x,y) is any point on ellipse, then
x=2+cosθ and y=13+13sinθ

4x9y=4(2+cosθ)9(13+13sinθ)
f(θ)=8+4cosθ+33sinθ
f(θ)=11+4cosθ3sinθ
f(θ)max=11+32+42=16
So maximum value of 4x9y2=162=8

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