If $|Z_1|=|Z_2|=|Z_3|=1$ and $Z_1+Z_2+Z_3=0$,then the area of the triangle whose vertices are $Z_1, Z_2, Z_3$ is

If $z$ is a complex number such that $z^2 = (\bar{z})^2$,then

If a point $P$ denotes the complex number $z=x+iy$ in the Argand plane and if $\frac{z-(2+i)}{z+(1-2i)}$ is purely real,then the locus of $P$ is

If $z_1$ and $z_2$ are two complex numbers satisfying the equation $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1$,then $\frac{z_1}{z_2}$ may be

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

If $|z_1| = |z_2|$ and $\arg\left( \frac{z_1}{z_2} \right) = \pi$,then $z_1 + z_2$ is equal to