If $\text{Im} \left( \frac{2z + 1}{iz + 1} \right) = -3$,then the locus of $z$ is :-

The figure in the complex plane given by $10 z \bar{z} - 3(z^2 + \bar{z}^2) + 4i(z^2 - \bar{z}^2) = 0$ is

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a triangle $ABC$ respectively in the Argand plane. If $|z_1-z_2|=\sqrt{25-12\sqrt{3}}$,$|\frac{z_1-z_3}{z_2-z_3}|=\frac{3}{4}$ and $\angle ACB=30^{\circ}$,then the area (in sq. units) of that triangle is

For any real number $r$, let $A_r = \{e^{i \pi r n} : n \in \mathbb{N}\}$ be a set of complex numbers. Then,

Let $z_{1}$ and $z_{2}$ be complex numbers such that $z_{1} \neq z_{2}$ and $|z_{1}|=|z_{2}|$. If $\operatorname{Re}(z_{1}) > 0$ and $\operatorname{Im}(z_{2}) < 0$, then $\frac{z_{1}+z_{2}}{z_{1}-z_{2}}$ is

$A$ man walks a distance of $3$ units from the origin towards the north-east $(N 45^{\circ} E)$ direction. From there,he walks a distance of $4$ units towards the north-west $(N 45^{\circ} W)$ direction to reach a point $P$. Then the position of $P$ in the Argand plane is