If $|z-2|=|z-1|$,where $z$ is a complex number,then the locus of $z$ is a straight line:

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 locus of a point $z$ satisfying $|z|^2 = \operatorname{Re}(z)$ is a circle with centre

The equation $z\overline{z} + a\overline{z} + \overline{a}z + b = 0$,where $b \in \mathbb{R}$,represents a circle if

Let the curve $z(1+i)+\bar{z}(1-i)=4, z \in \mathbb{C}$,divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals :

The centre of a regular polygon of $n$ sides is located at the point $z = 0$ and one of its vertices $z_1$ is known. If $z_2$ is the vertex adjacent to $z_1$,then $z_2$ is equal to