$A$ point $z$ moves in the complex plane such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{4}$,then the minimum value of $|z-9 \sqrt{2}-2 i|^{2}$ is equal to ..... .

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?
$(A)$ $|z-z_1| + |z-z_2| = |z_1-z_2|$
$(B)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z-z_2)$
$(C)$ $\left|\begin{array}{cc} z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1 \end{array}\right| = 0$
$(D)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z_2-z_1)$

If $z = \sqrt{2} - i\sqrt{2}$ is rotated through an angle $45^{\circ}$ in the anti-clockwise direction about the origin,then the coordinates of its new position are

If $Z \neq \pm 1$ is a complex number and $\operatorname{Arg}\left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$,then the locus of $Z$ in the Argand plane is

If $z = x + iy$ is a complex number and $|1 + iz| = |1 - iz|$, then

Let $a, b \in \mathbb{R}$ and $a^2+b^2 \neq 0$. Suppose $S = \{z \in \mathbb{C} : z = \frac{1}{a+ibt}, t \in \mathbb{R}, t \neq 0\}$,where $i = \sqrt{-1}$. If $z = x+iy$ and $z \in S$,then $(x, y)$ lies on: