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Question

If ampz1z+1=π3 then z represents a point on

A
a straight line
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B
a circle
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C
a pair of lines
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D
none of these
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Solution

The correct option is C a circle
|z1z+1|=k represents a circle if k>0 and k1
If k=1
Then
|z1z+1|=1
Represents a straight line perpendicular to x axis.
That is arg(z1z+1)=0
Hence for
arg(z1z+1)0 it will represent an arc of a circle.
Hence
arg(z1z+1)=π3 represents an arc of a circle.

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