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Question

If z=x+iy, x,yR and Im(2z+1i¯¯¯z+1)=2, then

A
4x2+6y+2+y2=0
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B
4y2+6y2+x=0
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C
4y2+6y+2x=0
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D
4y2+6y+2+x=0
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Solution

The correct option is C 4y2+6y+2x=0
Given : z=x+iy, so ¯¯¯z=xiy
Now,
2z+1i¯¯¯z+1=2x+1+2iyi(xiy)+1=(2x+1)+2iy(1+y)+ix×(1+y)ix(1+y)ix=(2x+1)(1+y)+2xy+i[2y(1+y)x(2x+1)](1+y)2+x2
So,
Im(2z+1i¯¯¯z+1)=22y(1+y)x(2x+1)(1+y)2+x2=22x2+x2y2y2=2y2+4y+2+2x24y2+6y+2x=0

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