$\sinh(ix)$ is equal to

Let $C$ be the circle in the complex plane with centre $z_0 = \frac{1}{2}(1 + 3i)$ and radius $r = 1$. Let $z_1 = 1 + i$ and the complex number $z_2$ be outside the circle $C$ such that $|z_1 - z_0| |z_2 - z_0| = 1$. If $z_0, z_1$ and $z_2$ are collinear,then the smaller value of $|z_2|^2$ is equal to $.............$.

If $z=x+iy$,then the equation $|z+1|=|z-1|$ represents

If $z$ is a complex number having the least absolute value and $|z - 2 + 2i| = 1$,then $z =$

The region represented by $\{z=x+iy \in \mathbb{C} : |z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality

For any two complex numbers $z_1, z_2$,if $|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2$,then: