If $z=x+iy$ is a complex number satisfying $\left|\frac{z-2i}{z+2i}\right|=2$ and the locus of $z$ is a circle,then its radius is

If $|z - 2 - 3i| + |z + 2 - 6i| = 4$,where $i = \sqrt{-1}$,then the locus of $P(z)$ is

If ${z_1}, {z_2}, {z_3}$ are three non-zero complex numbers such that ${z_2} \neq {z_1}$,$a = |{z_1}|$,$b = |{z_2}|$,and $c = |{z_3}|$. Suppose that $\left| {\begin{array}{*{20}{c}} a & b & c \\ b & c & a \\ c & a & b \end{array}} \right| = 0$,then $arg\left( {\frac{{{z_3}}}{{{z_2}}}} \right)$ is equal to:

The value of $|z|^2+|z-3|^2+|z-i|^2$ is minimum when $z$ equals

Let point $P = \alpha + i\beta$,where $\alpha, \beta > 0$,undergo the following three transformations successively on the Argand plane:
$(I)$ Reflection about $\text{amp}(z) = \frac{\pi}{4}$
$(II)$ Transformation through a distance $\beta$ units along the positive direction of the real axis
$(III)$ Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-clockwise direction
If the final position of the point is given by $Q = -\sqrt{2} + i\sqrt{6}$,then:

The locus of $z$ satisfying the inequality $\log_{1/3}|z + 1| > \log_{1/3}|z - 1|$ is