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

If $|{z_1}|\, = \,|{z_2}|$ and $arg\,\,\left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \pi $, then ${z_1} + {z_2}$ is equal to

$|{z_1} + {z_2}|\, = \,|{z_1}| + |{z_2}|$ is possible if

Let $\alpha=8-14 i , A=\left\{ z \in C : \frac{\alpha z -\bar{\alpha} \overline{ z }}{ z ^2-(\overline{ z })^2-112 i }=1\right\}$ and $B =\{ z \in C :| z +3 i |=4\}$ Then $\sum_{z \in A \cap B}(\operatorname{Re} z-\operatorname{Im} z)$ is equal to $...............$.

If ${z_1}$ and ${z_2}$ are two non-zero complex numbers such that $|{z_1} + {z_2}| = |{z_1}| + |{z_2}|,$then arg $({z_1}) - $arg $({z_2})$ is equal to

If $z_1 = 1+2i$ and $z_2 = 3+5i$ , then ${\mathop{\rm Re}\nolimits} \,\left( {\frac{{{{\overline Z }_2}{Z_1}}}{{{Z_2}}}} \right) = $