Question

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

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

$\arg \left(\frac{z-1}{z+i}\right)=\frac{\pi }{2}$
i.e., line segment joining ' $1$ ' and '$-i$' subtends right angle at variable point $P(z)$
Locus of point $P(z)$ is $C_1$ as shown in the figure.
Now, unimodoular complex numbers lie on the circle $C_2$ with center at origin and radius $1$.

Clearly, two point ' $1$ ' and ' $-i$ ' are possible points, but these points are not satisfying $(1)$.
Hence, no such complex number.