If the amplitude of $\frac{z-1}{z+1}$ is $\frac{\pi }{4}$, then the locus of $z$ is

Solution

1. The condition $\text{arg}(\frac{z-1}{z+1})=\frac{\pi }{4}$ represents the set of points $z$ such that the angle between vectors $z-1$ and $z+1$ is constant.
2. Geometrically, this is an arc of a circle passing through the points $1$ and $-1$.
3. The locus is a circular arc because the angle subtended by the chord connecting $1$ and $-1$ at any point on the locus is constant.
4. Hence the answer is (B).