If $z = x + iy$ and $z^2 = (i \bar{z})^2$,then

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 ......

If $|z-2+i| \leq 2$,then the difference between the greatest and least value of $|z|$ is $(i=\sqrt{-1})$.

Let $S_1 = \{z \in \mathbb{C} : |z| \leq 5\}$, $S_2 = \{z \in \mathbb{C} : \operatorname{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right) \geq 0\}$ and $S_3 = \{z \in \mathbb{C} : \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is:

Let $z$ be a complex number such that $\left|\frac{z-2i}{z+i}\right|=2$,where $z \neq -i$. Then $z$ lies on a circle of radius $2$ and center:

Let $a, b, c, d$ be real numbers such that $|a-b|=2$,$|b-c|=3$,and $|c-d|=4$. Then,the sum of all possible values of $|a-d|$ is