If a complex number $z$ statisfies the equation $x + \sqrt 2 \,\,\left| {z + 1} \right|\,+ \,i\, = \,0,$ then $\left| z \right|$ is equal to

If $0 < amp{\rm{ (z)}} < \pi {\rm{,}}$then $amp(z)-amp ( - z) = $

For $a \in C$, let $A =\{z \in C: \operatorname{Re}( a +\overline{ z }) > \operatorname{Im}(\bar{a}+z)\}$ and $B=\{z \in C: \operatorname{Re}(a+\bar{z}) < \operatorname{Im}(\bar{a}+z)\}$. Then among the two statements :

$(S 1)$ : If $\operatorname{Re}(A), \operatorname{Im}(A) > 0$, then the set $A$ contains all the real numbers

$(S2)$: If $\operatorname{Re}(A), \operatorname{Im}(A) < 0$, then the set $B$ contains all the real numbers,

Let $Z$ and $W$ be complex numbers such that $\left| Z \right| = \left| W \right|,$ and arg $Z$ denotes the principal argument of $Z.$

Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.

Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$

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 ${z_1}{\rm{ and }}{z_2}$ be complex numbers such that ${z_1} \ne {z_2}$ and $|{z_1}|\, = \,|{z_2}|$. If ${z_1}$ has positive real part and ${z_2}$ has negative imaginary part, then $\frac{{({z_1} + {z_2})}}{{({z_1} - {z_2})}}$may be