If the amplitude of $z-2-3i$ is $\pi/4$,then the locus of $z=x+iy$ is

If $\cos \alpha + \cos \beta + \cos \gamma = 0$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$,then $\sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma = $

If the point $\left(\frac{k-1}{k}, \frac{k-2}{k}\right)$ lies on the locus of $z$ satisfying the inequality $\left|\frac{z+3i}{3z+i}\right| < 1$,then the interval in which $k$ lies 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 a point $P$ denotes a complex number $z=x+iy$ in the Argand plane and if $\frac{z+1}{z+i}$ is a purely real number,then the locus of $P$ is

If $z = x + iy$ and $|z - 2 + i| = |z - 3 - i|$,then the locus of $z$ is: