The locus of $z$ satisfying the inequality $\log_{1/3}|z + 1| > \log_{1/3}|z - 1|$ is

If $|z+i|-|z-1|=|z|-2=0$ for a complex number $z$, then $z=$

If $z$ is a complex number such that $\frac{z-1}{z-i}$ is purely imaginary and the locus of $z$ represents a circle with centre $(\alpha, \beta)$ and radius $r$,then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=$

If ${z_1}, {z_2}, {z_3}, {z_4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|$,then ${z_1}, {z_2}, {z_3}, {z_4}$ are

The locus of the point $z$ satisfying $arg\left( \frac{z - 1}{z + 1} \right) = k$ (where $k$ is non-zero) is

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