Question

If $amp\left(\frac{z-1}{z+1}\right)=\frac{\pi }{3}$,Then z represent a point on

• A. a circle
• B. a straight line
• C. an ellipse
• D. a hyperbola.

Answer 1

The correct option is A a circle

$|\frac{z-1}{z+1}|=k$ represents a circle if $k>0$ and $k\ne 1$
If $k=1$
Then
$|\frac{z-1}{z+1}|=1$
Represents a straight line perpendicular to x axis.
That is $arg\left(\frac{z-1}{z+1}\right)=0$
Hence for
$arg\left(\frac{z-1}{z+1}\right)\ne 0$ it will represent an arc of a circle.
Hence
$arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{3}$ represents an arc of a circle.