If ${(\sqrt 8 + i)^{50}} = {3^{49}}(a + ib)$ then ${a^2} + {b^2}$ is

Let $z$ be a complex number such that $\left| z \right| + z = 3 + i$ (where $i = \sqrt { - 1} $). Then $\left| z \right|$ is equal to

${\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} - {z_2}} \right|^2}$ is equal to 

If $x+i y=\frac{a+i b}{a-i b},$ prove that $x^{2}+y^{2}=1$

Let $\mathrm{z}$ be a complex number such that $|\mathrm{z}+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is :

Argument of $ - 1 - i\sqrt 3 $ is