Let the curve $z(1+i)+\bar{z}(1-i)=4, z \in \mathbb{C}$,divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals :

Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i| \geq \sqrt{5}$. If the complex number $z_0$ is such that $\frac{1}{|z_0-1|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$,then the principal argument of $\frac{4-z_0-\bar{z}_0}{z_0-\bar{z}_0+2i}$ is

In the Argand plane,the values of $z$ satisfying the equation $|z-1|=|i(z+1)|$ lie on

If $a, b, c$ and $u, v, w$ are complex numbers representing the vertices of two triangles such that $c = (1 - r)a + rb$ and $w = (1 - r)u + rv$,where $r$ is a complex number,then the two triangles

Let $R$ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers,where $i = \sqrt{-1}$. Let $S = \{(x, y) \in R \times R : |x + iy - z_1| = 2|x + iy - z_2|\}$. Then which of the following statements is (are) True?
$(A) S$ is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$
$(B) S$ is a circle with centre $\left(\frac{1}{3}, \frac{8}{3}\right)$
$(C) S$ is a circle with radius $\frac{\sqrt{2}}{3}$
$(D) S$ is a circle with radius $\frac{2\sqrt{2}}{3}$

The equation of any $Circle$ in the complex plane is of the form $z \bar{z} + b \bar{z} + \bar{b} z + c = 0$,where $b \in \mathbb{C}$ and $c \in \mathbb{R}$.