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Question

If the imaginary part of 2z+1iz+1 is 2, then the locus of z is

A
a circle
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B
a straight line
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C
an ellipse
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D
a parabola
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Solution

The correct option is B a straight line
Let z=x+iy
2z+1iz+1=2(x+iy)+1i(x+iy)+1
=(2x+1)+2iy(1y)+ix×(1y)ix(1y)ix
=(2x+1)+2iy(1y)2+x2×{(1y)ix}
Imaginary part
=2y(1y)x(2x+1)(1y)2+x2=2
2y2y22x2x=2x22+4y2y2
x+2y2=0 which is a straight line.

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