If a complex number $z = x + iy$ is taken such that the amplitude of the fraction $\frac{z - 1}{z + 1}$ is always $\frac{\pi}{4}$,then:

The points in the set $\{z \in \mathbb{C} : \arg \left(\frac{z-2}{z-6i}\right) = \frac{\pi}{2}\}$ (where $\mathbb{C}$ denotes the set of all complex numbers) lie on the curve which is a

$A$ 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 $a, b \in \mathbb{R}$ 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:

If $\left| z - \frac{1 + 3i}{2} \right| = \frac{\sqrt{10}}{2}$ and $P$,$Q$,and $R$ are points representing the complex numbers $z$,$z e^{i \pi / 3}$,and $z(1 + e^{i \pi / 3})$ respectively in the Argand plane,then the area of the triangle $PQR$ is:

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 $.......$