$S = \{z \in \mathbb{C} : |z - 1 + i| = 1\}$ represents

$A$ particle $P$ starts from the point $z_0 = 1 + 2i$,where $i = \sqrt{-1}$. It moves first horizontally away from the origin by $5$ units and then vertically away from the origin by $3$ units to reach a point $z_1$. From $z_1$,the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{\pi}{2}$ in the anticlockwise direction on a circle with the center at the origin,to reach a point $z_2$. The point $z_2$ is given by:

Let $z_{1}$ and $z_{2}$ be two imaginary roots of $z^{2}+pz+q=0$,where $p$ and $q$ are real. The points $z_{1}, z_{2}$ and the origin form an equilateral triangle if

The locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$,where $z=x+iy$,is

If the amplitude of $z-2-3i$ is $\frac{\pi}{4}$,then the locus of $z=x+iy$ is:

If three complex numbers are in $A.P.$,then they lie on