Let the minimum value $v_{0}$ of $v = |z|^{2} + |z-3|^{2} + |z-6i|^{2}$,where $z \in \mathbb{C}$,be attained at $z = z_{0}$. Then $|2z_{0}^{2} - \bar{z}_{0}^{3} + 3|^{2} + v_{0}^{2}$ is equal to:

$A$ circle whose radius is $r$ and centre is $z_0$,then the equation of the circle is

Let $z_1$ and $z_2$ be two distinct complex numbers and let $z = (1-t)z_1 + tz_2$ for some real number $t$ with $0 < t < 1$. If $\operatorname{Arg}(w)$ denotes the principal argument of a non-zero complex number $w$,then which of the following are true?
$(A)$ $|z-z_1| + |z-z_2| = |z_1-z_2|$
$(B)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z-z_2)$
$(C)$ $\left|\begin{array}{cc} z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1 \end{array}\right| = 0$
$(D)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z_2-z_1)$

Let $z$ be a complex number such that $|z + 2| = |z - 2|$ and $\arg\left(\frac{z + 3}{z - i}\right) = \frac{\pi}{4}$. Then $|z|^2$ is equal to:

The values of $z$ for which $|z + i| = |z - i|$ are

The number of complex numbers $z$ such that $|z - 1| = |z + 1| = |z - i|$ is equal to