Let $z$ be a complex number such that $\left|\frac{z-2i}{z+i}\right|=2$,where $z \neq -i$. Then $z$ lies on a circle of radius $2$ and center:

If $P$ is the point in the Argand diagram corresponding to the complex number $\sqrt{3}+i$ and if $OPQ$ is an isosceles right-angled triangle,right-angled at $O$,then $Q$ represents the complex number

Let $z$ and $w$ be two non-zero complex numbers such that $|z| = |w|$ and $arg(z) + arg(w) = \pi$. Then $z$ is equal to:

The complex number $z = x + iy$ which satisfies the equation $\left| \frac{z - 5i}{z + 5i} \right| = 1$ lies on

The locus of the points $z$ which satisfy the condition $\text{arg} \left( \frac{z - 1}{z + 1} \right) = \frac{\pi}{3}$ is

If the eight vertices of a regular octagon are given by the complex numbers $z_j = \frac{1}{x_j - 2i}$ for $j = 1, 2, \dots, 8$,where $x_j$ are the roots of $x^8 - 1 = 0$,then the radius of the circumcircle of the octagon is