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

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 -

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of

$(\bar{z})^2+\frac{1}{z^2}$

are integers, then which of the following is/are possible value($s$) of $|z|$ ?

If $z_1, z_2  $ are any two complex numbers, then $|{z_1} + \sqrt {z_1^2 - z_2^2} |$ $ + |{z_1} - \sqrt {z_1^2 - z_2^2} |$ is equal to

If $|z|\, = 1,(z \ne - 1)$and $z = x + iy,$then $\left( {\frac{{z - 1}}{{z + 1}}} \right)$ is

If $5 + ix^3y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and arg $(x + iy) = \theta $ , then ${\tan ^2}\,\theta $ is equal to