The equation of any $Circle$ in the complex plane is of the form $z \bar{z} + b \bar{z} + \bar{b} z + c = 0$,where $b \in \mathbb{C}$ and $c \in \mathbb{R}$.

Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $X = \{ z \in \mathbb{C} : \operatorname{Re}(a z^2 + bz) = a \text{ and } \operatorname{Re}(b z^2 + az) = b \}$ is equal to

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 :

Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6i|=5$. Then the value of $z^{3}+3z^{2}-15z+141$ is equal to

If $|z + 4| \le 3$,then the greatest and the least value of $|z + 1|$ are

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