If a complex number $z$ satisfies $|z^2-1|=|z|^2+1$,then $z$ lies on

Let $S = \{z \in \mathbb{C} : z^{2} + \bar{z} = 0\}$. Then $\sum_{z \in S} (\operatorname{Re}(z) + \operatorname{Im}(z))$ is equal to $......$

If $z$ is a complex number such that $\frac{z - 1}{z + 1}$ is purely imaginary,then

The area of $\Delta$ whose vertices are $z, \omega z, z + \omega z$ is (where $\omega$ is a complex cube root of unity):

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$,then $3(\alpha+\beta+\gamma)$ is equal to

The points in the Argand plane given by $Z_1 = -3 + 5i$,$Z_2 = -1 + 6i$,$Z_3 = -2 + 8i$,and $Z_4 = -4 + 7i$ form a: