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Question

The image of the circle x2+y2=16 in the line x+y=4 is

A
x2+y24x4y+16=0
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B
x2+y2+4x4y+32=0
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C
x2+y28x8y+16=0
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D
x2+y28x8y+32=0
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Solution

The correct option is B x2+y28x8y+16=0
The parametric equation of the circle x2+y2=16 is x=4cosθ,y=4sinθ,θ being parameter so any point on the circle

x2+y2=16 is (4cosθ,4sinθ). Let (X,Y) be the image of the point (4cosθ,4sinθ) in the line x+y=4 then,

X4cosθ1=Y4sinθ1=2(4cosθ+4sinθ4)(12+12)

X4cosθ=Y4sinθ=(4cosθ+4sinθ4)

X4cosθ=4cosθ4sinθ+4 and Y4sinθ=4cosθ4sinθ+4

X4=4sinθ ....(i)

Y4=4cosθ ....(ii)

From (i) & (ii) we get

(X4)2+(Y4)2=16

X2+Y28X8Y+16=0
Hence choice (c) is correct answer..

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