The locus of the points represented by $|z+3|-|z-3|=6$,where $z$ is a complex number,is ....

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction,and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to $...........$.

If $|z + 4| \le 3$,then the maximum value of $|z + 1|$ is

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a triangle $ABC$ respectively in the Argand plane. If $|z_1-z_2|=\sqrt{25-12\sqrt{3}}$,$|\frac{z_1-z_3}{z_2-z_3}|=\frac{3}{4}$ and $\angle ACB=30^{\circ}$,then the area (in sq. units) of that triangle is

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 $\frac{2\sqrt{2}}{3}$

If $|Z_1|=|Z_2|=|Z_3|=1$ and $Z_1+Z_2+Z_3=0$, then the area of the triangle whose vertices are $Z_1, Z_2, Z_3$ is