The locus of the complex number $Z$ such that $\arg \left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$ is

The area of the triangle whose vertices are $i, \omega$ and $\omega^2$ is (Where $\omega$ is a complex cube root of unity other than $1$,$i$ is an imaginary number) . . . . . . sq.units

Let $A = \{ z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1 \}$ and $B = \{ z \in \mathbb{C} : \arg(\frac{z-1}{z+1}) = \frac{2\pi}{3} \}$. Then $A \cap B$ is

If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle,whose centroid is $z_0$,then $\sum_{k=1}^3 (z_k - z_0)^2$ is equal to

If $z = x + iy$,then the area of the triangle whose vertices are the points $z$,$iz$,and $z + iz$ is

If the center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1 + 2i$,then its perimeter is