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 $x+y=$

Let $z=x+iy$ be a complex number with $x, y \in \mathbb{Z}$. Then,the area (in sq units) of the rectangle whose vertices are the roots of the equation $\bar{z} \cdot z^3+z \cdot \bar{z}^3=350$ is

If ${z_1}, {z_2}, {z_3}$ are three non-zero complex numbers such that ${z_2} \neq {z_1}$,$a = |{z_1}|$,$b = |{z_2}|$,and $c = |{z_3}|$. Suppose that $\left| {\begin{array}{*{20}{c}} a & b & c \\ b & c & a \\ c & a & b \end{array}} \right| = 0$,then $arg\left( {\frac{{{z_3}}}{{{z_2}}}} \right)$ is equal to:

If $P$ is the point in the Argand diagram corresponding to the complex number $\sqrt{3}+i$ and if $OPQ$ is an isosceles right-angled triangle,right-angled at $O$,then $Q$ represents the complex number

If a complex number $z$ satisfies $|z|^2+1=|z^2-1|$,then the locus of $z$ is

Let $z \in \mathbb{C}$ be such that $|z| < 1$. If $w = \frac{5 + 3z}{5(1 - z)}$,then