The points representing the complex number $z$ for which $\text{arg}\left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ lie on

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

The vector $z = 3 - 4i$ is turned anticlockwise through an angle of $180^{\circ}$ and stretched $2.5$ times. The complex number corresponding to the newly obtained vector is

If $z$ is a complex number such that $z^2 = (\bar{z})^2$,then

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:

Let $\omega = z \bar{z} + k_1 z + k_2 i z + \lambda(1 + i)$,where $k_1, k_2 \in R$. Let $\operatorname{Re}(\omega) = 0$ be the circle $C$ of radius $1$ in the first quadrant touching the line $y = 1$ and the $y$-axis. If the curve $\operatorname{Im}(\omega) = 0$ intersects $C$ at $A$ and $B$,then $30(AB)^2$ is equal to $.......$.