If the area of the triangle formed by the points $z, z + iz$ and $iz$ on the complex plane is $18$,then the value of $|z|$ is

If $z=x+iy$,where $x$ and $y$ are real numbers and $i=\sqrt{-1}$,then the points $(x, y)$ for which $\frac{z-1}{z-i}$ is real,lie on

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

Let $S = \{z \in \mathbb{C} : z^{2} + \bar{z} = 0\}$. Then $\sum_{z \in S} (\operatorname{Re}(z) + \operatorname{Im}(z))$ is equal to $......$

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

The inequality $|z - 4| < |z - 2|$ represents the region given by