If the arguments of $(1-i)(\sqrt{3}+i)(1+\sqrt{3}i)$ and $(z-2)\left(\frac{-1}{z-1}\right)$ are equal, then the locus of $Z$ is part of a circle with centre $(a,b)$. The value of $\frac{a}{b+1}$ is
Step-by-Step Solution
Key Concept: The sum of $n$-th roots of unity equals zero, and the argument of quotients equals the difference of arguments
Given $1^n = \cos\frac{2\pi r}{n} + i\sin\frac{2\pi r}{n}$ for $r = 0, 1, \ldots, n-1$. Let $z_1 = \cos\frac{2\pi r}{n} + i\sin\frac{2\pi r}{n}$ and $z_2 = \cos\frac{2\pi r}{n} + i\sin\frac{2\pi r}{n}$. Then, $\sum z_i Q_2 = \text{amp}\left(\frac{r}{r}\right) = \text{amp}(z_1) - \text{amp}(z_2) = \frac{2(n-1)n}{2} = \frac{2}{n}$ (Given). Therefore, $n = 4(r_1 - r_2) = 4s$, so $n$ is of the form $4k$.
Correct Answer: 4