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

Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius $2$. Let $P$ be a point on the circle and let $PA_i$ denote the distance between the points $P$ and $A_i$ for $i=1, 2, \ldots, 8$. If $P$ varies over the circle,then the maximum value of the product $PA_1 \cdot PA_2 \cdot \cdots \cdot PA_8$ is:

If $A = \{z = x + iy : \text{real part of } \frac{\bar{z}-1}{z-i} = 2\}$,then the locus of the point $P(x, y)$ in the Cartesian plane is:

The set of all $\alpha \in R$,for which $w = \frac{1 + (1 - 8\alpha)z}{1 - z}$ is a purely imaginary number,for all $z \in C$ satisfying $|z| = 1$ and $\text{Re}(z) \neq 1$,is

If a complex number $z$ is such that $\frac{z-2i}{z-2}$ is a purely imaginary number and the locus of $z$ is a closed curve,then the area of the region bounded by that closed curve and lying in the first quadrant is

The area of the triangle whose vertices are complex numbers $z, iz, z + iz$ in the Argand diagram is