If $z_1 = 1 + 2i$,$z_2 = 2 + 3i$,and $z_3 = 3 + 4i$,then $z_1, z_2, z_3$ represent the vertices of a/an:

If $w = \frac{z}{z - \frac{1}{3}i}$ and $|w| = 1$,where $i = \sqrt{-1}$,then $z$ lies on

If ${z_1}, {z_2}, {z_3}$ are affixes of the vertices $A, B$ and $C$ respectively of a triangle $ABC$ having centroid at $G$ such that $z = 0$ is the midpoint of $AG$,then:

If a complex number $z$ satisfies $|z|^2+1=|z^2-1|$,then the locus of $z$ is

The sum of the square of the modulus of the elements in the set $\{z=a+ib: a, b \in \mathbb{Z}, z \in \mathbb{C}, |z-1| \leq 1, |z-5| \leq |z-5i|\}$ is ........

Let $S_1 = \{z \in \mathbb{C} : |z| \leq 5\}$, $S_2 = \{z \in \mathbb{C} : \operatorname{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right) \geq 0\}$ and $S_3 = \{z \in \mathbb{C} : \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is: