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

Let $z = x + iy$, where $x$ and $y$ are real. The points $(x, y)$ in the $X-Y$ plane for which $\frac{z+i}{z-i}$ is purely imaginary, lie on

Let $z_{1}$ be a fixed point on the circle of radius $1$ centered at the origin in the Argand plane and $z_{1} \neq \pm 1$. Consider an equilateral triangle inscribed in the circle with $z_{1}, z_{2}, z_{3}$ as the vertices. Then, $z_{1} z_{2} z_{3}$ is equal to

The equation $|z - 5i| / |z + 5i| = 12,$ where $z = x + iy,$ represents a/an

Let $z=x+iy$ represent a point $P(x, y)$ in the Argand plane. If $z$ satisfies the condition that $\text{arg}\left(\frac{z-3}{z-2i}\right)=-\frac{\pi}{2}$,then the locus of $P$ is

The number of solutions for the equations $|z - 1| = |z - 2| = |z - i|$ is