If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3i, 2+5i, 3i$ lie on a circle,then the centre of the circle is

The locus of $z$ satisfying $|z|+|z-1|=3$ is

Let $z_1, z_2, z_3, \omega, z_0, z'_0$ be fixed points on the complex plane such that no $3$ are collinear,satisfying the condition $Arg\left( \frac{\omega - z_1}{z_2 - z_3} \right) = Arg\left( \frac{\omega - z_2}{z_3 - z_1} \right) = Arg\left( \frac{\omega - z_3}{z_1 - z_2} \right) = \frac{\pi}{2}$. If $z_1, z_2, z_3$ satisfy the equation $|z - z_0| = R_1$ and $z_2, \omega, z_3$ satisfy the equation $|z - z'_0| = R_2$,then the ratio $\frac{R_1}{R_2}$ is equal to:

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:

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

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