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 .... .

The area of the triangle whose vertices are represented by the complex numbers $0, z,$ and $z{e^{i\alpha }}$ $(0 < \alpha < \pi )$ is equal to:

If $z$ is a complex number such that $\frac{z-1}{z-i}$ is purely imaginary and the locus of $z$ represents a circle with centre $(\alpha, \beta)$ and radius $r$,then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=$

For any integer $k$,let $\alpha_k = \cos \left(\frac{k \pi}{7}\right) + i \sin \left(\frac{k \pi}{7}\right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^{12} |\alpha_{k+1} - \alpha_k|}{\sum_{k=1}^3 |\alpha_{4k-1} - \alpha_{4k-2}|}$ is

The locus of the point $z=x+iy$ satisfying the equation $\left|\frac{z-1}{z+1}\right|=1$ is given by :

In the complex plane $\mathbb{C}$,the set $\{z \in \mathbb{C} : \arg \left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}\}$ represents