If $z_1, z_2, z_3$ are the vertices of an equilateral triangle and $z$ is its circumcentre,then

If the complex numbers $z_1, z_2, z_3$ represent the vertices of an equilateral triangle such that $|z_1| = |z_2| = |z_3|$,then $z_1 + z_2 + z_3 = $

Let $S = \{z \in \mathbb{C} : |z-1|=1 \text{ and } (\sqrt{2}-1)(z+\bar{z}) - i(z-\bar{z}) = 2\sqrt{2}\}$. Let $z_1, z_2 \in S$ be such that $|z_1| = \max_{z \in S} |z|$ and $|z_2| = \min_{z \in S} |z|$. Then $|\sqrt{2}z_1 - z_2|^2$ equals:

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

$A$ man walks a distance of $3$ units from the origin towards the north-east $(N 45^{\circ} E)$ direction. From there,he walks a distance of $4$ units towards the north-west $(N 45^{\circ} W)$ direction to reach a point $P$. Then the position of $P$ in the Argand plane is

If the imaginary part of $\frac{2z + 1}{iz + 1}$ is $-2$,then the locus of the point representing $z$ in the complex plane is