If $Z \neq \pm 1$ is a complex number and $\operatorname{Arg}\left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$,then the locus of $Z$ in the Argand plane is

The locus of the complex number $z$ such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ is:

Let $S = \{z \in \mathbb{C} : \left|\frac{z-6i}{z-2i}\right| = 1 \text{ and } \left|\frac{z-8+2i}{z+2i}\right| = \frac{3}{5}\}$. Then $\sum_{z \in S} |z|^2$ is equal to

If $|z|=1$ and $w=\frac{z-1}{z+1}$ (where $z \neq -1$),then $\operatorname{Re}(w)$ is

Let $S = \{z \in \mathbb{C} - \{i, 2i\} : \frac{z^2 + 8iz - 15}{z^2 - 3iz - 2} \in \mathbb{R} \}$. If $\alpha - \frac{13}{11}i \in S$ and $\alpha \in \mathbb{R} - \{0\}$,then $242\alpha^2$ is equal to

Let $S = \{ z \in \mathbb{C} : |z - 2| \leq 1, z(1 + i) + \overline{z}(1 - i) \leq 2 \}$. Let $|z - 4i|$ attain minimum and maximum values,respectively,at $z_1 \in S$ and $z_2 \in S$. If $5(|z_1|^2 + |z_2|^2) = \alpha + \beta \sqrt{5}$,where $\alpha$ and $\beta$ are integers,then the value of $\alpha + \beta$ is equal to