In the complex plane $\mathbb{C}$,the set $\{z \in \mathbb{C} : \arg \left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}\}$ represents

If the center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1 + 2i$,then its perimeter is

The figure in the complex plane given by $10 z \bar{z} - 3(z^2 + \bar{z}^2) + 4i(z^2 - \bar{z}^2) = 0$ is

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4i}\right)=0$ and $\operatorname{Im}\left(\frac{z-2}{z-4i}\right)=1$ simultaneously is

$z_1$ and $z_2$ are two fixed points on the Argand plane. If $z$ is a complex number such that $|z-z_1| + |z-z_2| = \lambda$,then the locus of $z$ is

Let $z$ be a complex number such that $|z + 2| = |z - 2|$ and $\arg\left(\frac{z + 3}{z - i}\right) = \frac{\pi}{4}$. Then $|z|^2$ is equal to: