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

$PQ$ and $PR$ are two infinite rays. $QAR$ is an arc. $A$ point lying in the shaded region,excluding the boundary,satisfies:

Let $S = \{z : 3 \le |2z - 3(1 + i)| \le 7\}$ be a set of complex numbers. Then $\min_{z \in S} |z + \frac{1}{2}(5 + 3i)|$ is equal to:

If the eight vertices of a regular octagon are given by the complex numbers $z_j = \frac{1}{x_j - 2i}$ for $j = 1, 2, \dots, 8$,where $x_j$ are the roots of $x^8 - 1 = 0$,then the radius of the circumcircle of the octagon is

If $|8 + z| + |z - 8| = 16$,where $z$ is a complex number,then the point $z$ will lie on

Let the complex numbers $\alpha$ and $\left(\frac{1}{\bar{\alpha}}\right)$ lie on circles $\left(x-x_0\right)^2+\left(y-y_0\right)^2=r^2$ and $\left(x-x_0\right)^2+\left(y-y_0\right)^2=4 r^2$ respectively. If $z_0=x_0+i y_0$ satisfies the equation $2|z_0|^2=r^2+2$,then $|\alpha|=$