Let $z_1 = 6 + i$ and $z_2 = 4 -3i$. Let $z$ be a complex number such that $arg\ \left( {\frac{{z - {z_1}}}{{{z_2} - z}}} \right) = \frac{\pi }{2}$, then $z$ satisfies -

Argument and modulus of $\frac{{1 + i}}{{1 - i}}$ are respectively

Let $z$ and $w$ be two complex numbers such that $w=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1$ and $\operatorname{Re}(w)$ has minimum value. Then, the minimum value of $n \in N$ for which $w ^{ n }$ is real, is equal to..........

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are $TRUE$?

$(A)$ $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$  $(B)$ $|z| \leq 2$ for all $z \in S$

$(C)$ $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$  $(D)$ The set $S$ has exactly four elements

If $z=\frac{1}{2}-2 i$, is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in R \quad$, then $\alpha+\beta$ is equal to

Find the modulus and the argument of the complex number $z=-1-i \sqrt{3}$.