For $z \in \mathbb{C}$,if the minimum value of $(|z-3 \sqrt{2}| + |z-p \sqrt{2} i|)$ is $5 \sqrt{2}$,then a value of $p$ is $.......$

If the center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1 + 2i$,then its perimeter 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 the locus of a point $z$ in the Argand plane satisfying the condition $\operatorname{Re}(z^2)=4$ be $C_1$ and the locus of $z$ satisfying the condition $\operatorname{Im}(z^2)=4$ be $C_2$. Then the number of common points of the two curves $C_1$ and $C_2$ are

Let $z_{1}$ and $z_{2}$ be the roots of the equation $z^{2} + az + 12 = 0$. If $z_{1}$,$z_{2}$,and the origin form an equilateral triangle in the complex plane,then the value of $|a|$ is:

If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle,whose centroid is $z_0$,then $\sum_{k=1}^3 (z_k - z_0)^2$ is equal to