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Q.
Number of unimodular complex number which satisfies the locus $\arg \left(\frac{z-1}{z+i}\right)=\frac{\pi}{2}$ is
Complex Numbers and Quadratic Equations
Solution:
$\arg \left(\frac{z-1}{z+i}\right)=\frac{\pi}{2}$
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.
