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Question

If Arg(z+1z1)=π6 , then find the locus of z.

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Solution

Given,

z+1z1

=x+iy+1x+iy1

=(x+1)+iy(x1)+iy×(x1)iy(x1)iy

=(x+1)(x1)+y22iy(x1)2+y2

z=x21+y2(x1)2+y2i2y(x1)2+y2

θ=tan1(ba)=tan1(2yx2+y21)

tanπ6=13

13=2yx2+y21

x2+y21=23y

x2+y2+0x23y1=0

g=0,f=3,c=1

center=(g,f)=(0,3)

radius=g2+f2c=02+(3)2(1)=2

Locus of z is a circle with center (0,3) and radius 2

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