If for the complex numbers $z$ satisfying $|z-2-2 i| \leq 1$,the maximum value of $|3 i z+6|$ is attained at $a+i b$,then $a+b$ is equal to .... .

Let a complex number be $w = 1 - \sqrt{3} i$. Let another complex number $z$ be such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$. Then the area of the triangle with vertices at the origin,$z$,and $w$ is equal to ........ .

The locus of the point representing the complex number $z$ for which $|z+3|^2-|z-3|^2=15$ is

If the equation $a|z|^2 + \overline{\bar{\alpha}z + \alpha\bar{z}} + d = 0$ represents a circle where $a, d$ are real constants,then which of the following conditions is correct?

Let $S = \{z = x + iy : \frac{2z - 3i}{4z + 2i} \text{ is a real number} \}$. Then which of the following is $NOT$ correct?

Let the point $P$ represent $z=x+iy$, where $x, y \in \mathbb{R}$, in the Argand plane. Let the curves $C_1$ and $C_2$ be the loci of $P$ satisfying the conditions $(i)$ $\frac{2z+i}{z-2}$ is purely imaginary and $(ii)$ $\operatorname{Arg}\left(\frac{z+i}{z+1}\right)=\frac{\pi}{2}$, respectively. Then the point of intersection of the curves $C_1$ and $C_2$, other than the origin, is