If the imaginary part of $\frac{2 z+1}{i z+1}$ is $-2$,then the locus of the point representing $z$ in the complex plane is

Let $z = x + iy$, where $x$ and $y$ are real. The points $(x, y)$ in the $X-Y$ plane for which $\frac{z+i}{z-i}$ is purely imaginary, lie on

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

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

If $z_1=2+3i$,$z_2=4-5i$,and $z_3$ are three points in the Argand plane such that $5z_1+xz_2+yz_3=0$ $(x, y \in R)$ and $z_3$ is the midpoint of the segment joining the points $z_1$ and $z_2$,then $x+y=$

If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3i, 2+5i, 3i$ lie on a circle,then the centre of the circle is