For the real parameter $t$, the locus of the complex number $z = (1 - t^2) + i \sqrt{1 + t^2}$ in the complex plane is

Let $A = \{z \in \mathbb{C} : 1 \leq |z - (1 + i)| \leq 2\}$ and $B = \{z \in A : |z - (1 - i)| = 1\}$. Then,$B$ is:

If $Z$ is a complex number such that $|Z| \leq 3$ and $-\frac{\pi}{2} \leq \operatorname{amp}(Z) \leq \frac{\pi}{2}$,then the area of the region formed by the locus of $Z$ is

The area of the triangle with vertices $A(z)$,$B(iz)$,and $C(z+iz)$ is

Let $S=S_1 \cap S_2 \cap S_3$,where $S_1=\{z \in \mathbb{C}:|z|<4\}$,$S_2=\{z \in \mathbb{C}: \operatorname{Im}[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}]>0\}$,and $S_3=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$.
$1.$ Area of $S=$
$(A) \frac{10 \pi}{3} \quad (B) \frac{20 \pi}{3} \quad (C) \frac{16 \pi}{3} \quad (D) \frac{32 \pi}{3}$
$2.$ $\min _{z \in S}|1-3 i-z|=$
$(A) \frac{2-\sqrt{3}}{2} \quad (B) \frac{2+\sqrt{3}}{2} \quad (C) \frac{3-\sqrt{3}}{2} \quad (D) \frac{3+\sqrt{3}}{2}$

Let $z = x + iy$ be a point in the Argand plane. If the amplitude of $\left(\frac{z - 3}{z + 2i}\right)$ is $\frac{\pi}{2}$,then the locus of $z$ is