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Question

The coordinates of the point on the circle x2+y2−2x−4y−11=0 farthest from the origin are

A
(2+85,1+45)
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B
(1+45,2+85)
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C
(1+85,2+45)
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D
None of these
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Solution

The correct option is B (1+4√5,2+8√5)The equation of the given circle can be written as (x−1)2+(y−2)2=16=42So, the coordinates of any point P on the circle are (1+4cosθ,2+4sinθ) whose distance from the origin is d=√(1+4cosθ)2+(2+4sinθ)2=√21+8cosθ+16sinθ⇒d=√21+8(cosθ+2sinθ)=√21+8rcos(θ−α)where rcosα=1,rsinα=2=√21+8√5cos(θ−α)which is maximum when cos(θ−α)=1.i.e., θ=α=tan−12.⇒tanθ=2 so sinθ=2√5 and cosθ=1√5 and, the coordinates of the required point are (1+4√5,2+8√5)

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