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

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a triangle $ABC$ respectively in the Argand plane. If $|z_1-z_2|=\sqrt{25-12\sqrt{3}}$,$|\frac{z_1-z_3}{z_2-z_3}|=\frac{3}{4}$ and $\angle ACB=30^{\circ}$,then the area (in sq. units) of that triangle is

If a complex number $z = x + iy$ is taken such that the amplitude of the fraction $\frac{z - 1}{z + 1}$ is always $\frac{\pi}{4}$,then:

The maximum distance from the origin of coordinates to the point $z$ satisfying the equation $\left| z + \frac{1}{z} \right| = a$ is

The locus of $z$ satisfying the inequality $\left|\frac{z+2 i}{2 z+i}\right| < 1$,where $z=x+i y$,is

$z_1$ and $z_2$ are the roots of $3z^2 + 3z + b = 0$. If the origin,$A(z_1)$,and $B(z_2)$ form an equilateral triangle,then the value of $b$ is equal to