The points $P$ and $Q$ denote the complex numbers $Z_1$ and $Z_2$ in the Argand plane. $O$ is the origin. If $Z_1 \bar{Z}_2 + \bar{Z}_1 Z_2 = 0$ and $\angle POQ = \theta$,then $\sin \theta = $

Let complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$,respectively. If $z_0=x_0+iy_0$ satisfies the equation $2|z_0|^2=r^2+2$,then $|\alpha|=$

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = \{w^n : n = 1, 2, 3, \ldots\}$. Further, $H_1 = \{z \in C : \operatorname{Re}(z) > \frac{1}{2}\}$ and $H_2 = \{z \in C : \operatorname{Re}(z) < -\frac{1}{2}\}$, where $C$ is the set of all complex numbers. If $z_1 \in P \cap H_1$, $z_2 \in P \cap H_2$, and $O$ represents the origin, then $\angle z_1 O z_2$ can be:

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 $A_1, A_2, A_3, \ldots, A_8$ 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:

The area of $\Delta$ whose vertices are $z, \omega z, z + \omega z$ is (where $\omega$ is a complex cube root of unity):