The locus of the point representing the complex number $z$ for which $|z+3|^2-|z-3|^2=15$ is

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:

If $z_1$ and $z_2$ are two distinct complex numbers such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$,then:

The area (in sq units) of the triangle whose vertices are the points represented by the complex numbers $0, z$,and $z e^{i \alpha}$ $(0 < \alpha < \pi)$ is:

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 $z$ is a complex number satisfying $|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=4,$ then $|z|$ cannot be