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Question

Let z=x+iy be a complex number. If 1+¯¯¯zz is a real number, then

A
x=12 or y=12
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B
x=12 or y=1
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C
x=0 or y=12
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D
x=12 or y=0
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Solution

The correct option is D x=12 or y=0
Given : z=x+iy, then
¯¯¯z=xiy
Now,
1+¯¯¯zz=1+xiyx+iy=1+xiyx+iy×xiyxiy=x(1+x)y2+i(2xyy)x2+y2
As 1+¯¯¯zz is a real number, so
2xyyx2+y2=0y(2x+1)=0y=0 or x=12

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