If ${z_1} = 10 + 6i,{z_2} = 4 + 6i$ and $z$ is a complex number such that $amp\left( {\frac{{z – {z_1}}}{{z – {z_2}}}} \right) = \frac{\pi }{4},$ then the value of $|z – 7 – 9i|$ is equal to

If ${z_1}.{z_2}……..{z_n} = z,$ then $arg\,{z_1} + arg\,{z_2} + ….$+$arg\,{z_n}$ and $arg$$z$ differ by a

If $\bar z$ be the conjugate of the complex number $z$, then which of the following relations is false

If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $arg(z) -arg(w) =\frac {\pi }{2},$ then

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