Let $z$ be a complex number such that $|z + 2| = |z - 2|$ and $\arg\left(\frac{z + 3}{z - i}\right) = \frac{\pi}{4}$. Then $|z|^2$ is equal to:

The set of points on the Argand plane which satisfy both $|z| \leq 4$ and $\operatorname{Arg}(z) = \frac{\pi}{3}$ represents:

Let $C$ be the set of all complex numbers. Let $S_{1} = \{z \in C : |z-3-2i|^{2}=8\}$,$S_{2} = \{z \in C : \operatorname{Re}(z) \geq 5\}$,and $S_{3} = \{z \in C : |z-\bar{z}| \geq 8\}$. Then the number of elements in $S_{1} \cap S_{2} \cap S_{3}$ is equal to:

If the amplitude of $z-2-3i$ is $\frac{\pi}{4}$,then the locus of $z=x+iy$ is:

Let $z$ be a complex number satisfying $|z+5| \leq 4$ and $z(1+i)+\bar{z}(1-i) \geq -10$,where $i=\sqrt{-1}$. If the maximum value of $|z+1|^2$ is $\alpha+\beta \sqrt{2}$,then the value of $(\alpha+\beta)$ is ......

Geometrically,the set $\{z \in \mathbb{C} : |z - 2 - 2i| \leq 1\}$ represents