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

The complex numbers $z_1, z_2$ and $z_3$ satisfying $\frac{z_1 - z_3}{z_2 - z_3} = \frac{1 - i\sqrt{3}}{2}$ are the vertices of a triangle which 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 complex number $z = x + iy$ which satisfies the equation $\left| \frac{z - 5i}{z + 5i} \right| = 1$ lies on

If $|z|=1$ and $z \neq \pm 1$,then all the values of $\frac{z}{1-z^2}$ lie on

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$,if $\sqrt{\lambda - 1}$ is the radius of the circle $|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$,then $|\alpha - \beta|$ is equal to $.............$.