If $z_1=2+3i$,$z_2=4-5i$,and $z_3$ are three points in the Argand plane such that $5z_1+xz_2+yz_3=0$ $(x, y \in R)$ and $z_3$ is the midpoint of the segment joining the points $z_1$ and $z_2$,then $x+y=$

If the complex numbers $z_1, z_2, z_3$ represent the vertices of an equilateral triangle such that $|z_1| = |z_2| = |z_3|$,then $z_1 + z_2 + z_3 = $

The polar coordinates of the point,whose Cartesian coordinates are $(-2 \sqrt{3}, 2)$,are

If $w = \frac{z}{z - \frac{1}{3}i}$ and $|w| = 1$,where $i = \sqrt{-1}$,then $z$ lies on

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = \{w^n : n = 1, 2, 3, \ldots\}$. Further, $H_1 = \{z \in C : \operatorname{Re}(z) > \frac{1}{2}\}$ and $H_2 = \{z \in C : \operatorname{Re}(z) < -\frac{1}{2}\}$, where $C$ is the set of all complex numbers. If $z_1 \in P \cap H_1$, $z_2 \in P \cap H_2$, and $O$ represents the origin, then $\angle z_1 O z_2$ can be:

The equation $z\overline{z} + (2 - 3i)z + (2 + 3i)\overline{z} + 4 = 0$ represents a circle of radius