If $z_1, z_2, z_3$ are vertices of a triangle in the Argand plane such that $|z_1 - z_2| = |z_1 - z_3|$,then $\arg \left( \frac{2z_1 - z_2 - z_3}{z_3 - z_2} \right)$ is:

The locus of the points $z$ which satisfy the condition $\text{arg} \left( \frac{z - 1}{z + 1} \right) = \frac{\pi}{3}$ is

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

If $\omega_1$ and $\omega_2$ are two non-zero complex numbers and $a, b$ are non-zero real numbers such that $|a \omega_1 + b \omega_2| = |a \omega_1 - b \omega_2|$,then $\frac{\omega_1}{\omega_2}$ is

Suppose that $z_{1}, z_{2}, z_{3}$ are three vertices of an equilateral triangle in the Argand plane. Let $\alpha = \frac{1}{2}(\sqrt{3} + i)$ and $\beta$ be a non-zero complex number. The points $\alpha z_{1} + \beta, \alpha z_{2} + \beta, \alpha z_{3} + \beta$ will be

$\sinh(ix)$ is equal to: