Question

If $arg\frac{z-1}{z+1}=\frac{\pi }{2}$ then the locus of $z$ is:

• A. Circle with radius $2$
• B. Circle with radius $1$
• C. Srtight line
• D. Pair of lines

Answer 1

The correct option is B Circle with radius $1$

$arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{2}implies$
angle between the complex numbers
$z-1$ and $z+1$ is $\frac{\pi }{2}$ in complex plane.

Clearly, as angle subtended by a diametric at point on circle is ${90}^0$
$\Rightarrow$ Here -1 and 1 are end-points of a diameter.
$\Rightarrow$ diameter =$|1-(-1\left)\right|=2$
radius =$1$ unit