The locus represented by $|z - 1| = |z + i|$ is

Suppose $z \in \mathbb{C}$ has an argument $\theta$ such that $0 < \theta < \frac{\pi}{2}$ and satisfies the equation $|z - 3i| = 3$. What is the value of $\cot \theta - \frac{6}{z}$?

$\sinh(ix)$ is equal to

Let $\alpha = 8 - 14i$,$A = \{ z \in \mathbb{C} : \frac{\alpha z - \bar{\alpha} \bar{z}}{z^2 - (\bar{z})^2 - 112i} = 1 \}$,and $B = \{ z \in \mathbb{C} : |z + 3i| = 4 \}$. Then $\sum_{z \in A \cap B} (\operatorname{Re}(z) - \operatorname{Im}(z))$ 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:

Let $z=x+iy$ and $P(x, y)$ be a point on the Argand plane. If $z$ satisfies the condition $\operatorname{Arg}\left(\frac{z-3i}{z+2i}\right)=\frac{\pi}{4}$, then the locus of $P$ is: