If z complex number satisfying `|z-1| = 1`, then which of the following is correct

Options

A. `"arg"(z-1)=2"arg"(z)`
B. `2arg(z) =2/3"arg"(z^(2)-z)`
C. `"arg"(z-1)=2"arg"(z+1)`
D. `"arg"z=2"arg"(z+1)`

Answer

A

Explanation

Clearly, `|z-1|=1` represents a circle with center at (1,0) and radius 1. Let P(z) be any point on it. Then,
`"arg"(z-1)=angleXCP= theta` (say)
`therefore "arg"(z) = angleXOP = theta/2`
Hence, arg(z-1) = 2arg(z)