The locus represented by $|z - 1| = |z + i|$ is

Let $z=x+iy$ and a point $P$ represent $z$ in the Argand plane. If the real part of $\frac{z-1}{z+i}$ is $1$,then a point that lies on the locus of $P$ is

$z_1, z_2, z_3$ 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

The area of the triangle whose vertices are complex numbers $z, iz, z + iz$ in the Argand diagram is

If a complex number $z$ satisfies $|z|^2+1=|z^2-1|$,then the locus of $z$ is

If $|z| = 1$ $(z \neq -1)$ and $z = x + iy$,then $\left( \frac{z - 1}{z + 1} \right)$ is