Question

Number of unimodular complex number which satisfies the locus $arg\left(\frac{z-1}{z+i}\right)=\frac{\pi }{2}$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

Solution:- (A) 0

$arg\left(\frac{z-1}{z+i}\right)=\frac{\pi }{2}$

Line segment joining '$1$' and '$-i$' subtends right angle at varia]ble point $P\left(z\right)$.

Locus of point $P\left(z\right)$ is $C_1$ as shown in the attached figure.

Now, unimodular complex numbers lie on the circle $C_2$ with centre at the origin, i.e., $(0,0)$ and radius $1$.

Clearly, $1$ and $-i$ are the two possible points but these points are not satisfying equation $\left(1\right)$

Hence no such complex number exist.

Hence the required answer is (A) 0.