If $a$ is a complex number and $b$ is a real number,then the equation $\bar{a}+a+b=0$ represents $a$ as a:

The points $P$ and $Q$ denote the complex numbers $Z_1$ and $Z_2$ in the Argand plane. $O$ is the origin. If $Z_1 \bar{Z}_2 + \bar{Z}_1 Z_2 = 0$ and $\angle POQ = \theta$,then $\sin \theta = $

Let $A = \{z \in \mathbb{C} : 1 \leq |z - (1 + i)| \leq 2\}$ and $B = \{z \in A : |z - (1 - i)| = 1\}$. Then,$B$ is:

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

If the point $P$ represents the complex number $z=x+iy$ in the Argand plane and if $\frac{z+i}{z-1}$ is a purely imaginary number, then the locus of $P$ is:

For $a, b, c, d \in R$, if $z_1 = a + ib$ and $z_2 = c + id$ are such that $|z_1| = |z_2| = 1$ and $\operatorname{Re}(z_1 \bar{z}_2) = 0$, then the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfy