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Question

P represents the variable complex number z. Find the locus of P, if Im[2z+iiz1]=1

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Solution

Let z=x+iy
2z+iiz1=2(x+iy)i(x+iy)1=2x+i(1+2y)y1+ix
=2x+i(1+2y)×(1+y)=ix(1+y)+ix×(1+y)ix
Given, Im=[2z+iiz1]=1
2x2(1+2y)(1+y)[(1+y)]2+x2=1
2x2(1+y+2y+2y2)1+y2+2y+x2=1
2x213y2y2=1y22yx2
x2+y2+y=0
locus of the P is therefore x2+y2+y=0.

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