In the complex plane $\mathbb{C}$,the set $\{z \in \mathbb{C} : \arg \left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}\}$ represents

The number of complex numbers $z$ such that $|z - 1| = |z + 1| = |z - i|$ is equal to

The equation $|z+1-i|=|z-1+i|$ represents a (where $z$ is a complex number)

The locus of the point $z=x+iy$ satisfying $\left|\frac{z-2i}{z+2i}\right|=1$ is

Which of the following is correct?

$A$ 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 upwards parallel to the positive $y$-axis by $3$ units to reach a point $Z_1$. From $Z_1$,the particle moves $\sqrt{2}$ units in the direction of vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{\pi}{2}$ in the anticlockwise direction on a circle with the centre at the origin to reach point $Z_2$. Then $Z_2 =$