Let $z = x + iy$ be a non-zero complex number such that $z^{2} = i|z|^{2},$ where $i = \sqrt{-1}.$ Then $z$ lies on the:

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4i}\right)=0$ and $\operatorname{Im}\left(\frac{z-2}{z-4i}\right)=1$ simultaneously is

The points $P$ and $Q$ denote the complex numbers $Z_1$ and $Z_2$ in the Argand plane. $O$ is the origin. If $Z_1 \bar{Z}_2 + \bar{Z}_1 Z_2 = 0$ and $\angle POQ = \theta$,then $\sin \theta = $

The maximum distance from the origin of coordinates to the point $z$ satisfying the equation $\left| z + \frac{1}{z} \right| = a$ is

The complex numbers $z_1, z_2$ and $z_3$ satisfying $\frac{z_1 - z_3}{z_2 - z_3} = \frac{1 - i\sqrt{3}}{2}$ are the vertices of a triangle which is

If $z = x + iy$ and $z^2 = (i \bar{z})^2$,then