The locus of the complex number $z$ such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ is:

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

$A$ 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:

For any complex number $z$, the minimum value of $|z| + |z - 1|$ is

If $C$ is a point on the straight line joining the points $A(-2+i)$ and $B(3-4i)$ in the Argand plane and $\frac{AC}{CB}=\frac{1}{2}$,then the argument of $C$ is

The locus of the point representing the complex number $z$ for which $|z+3|^2-|z-3|^2=15$ is