For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$,if $\sqrt{\lambda - 1}$ is the radius of the circle $|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$,then $|\alpha - \beta|$ is equal to $.............$.

If the four complex numbers $z$,$\overline{z}$,$\overline{z}-2 \operatorname{Re}(\overline{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side $4$ units in the Argand plane,then $|z|$ is equal to:

For all complex numbers $z_1$ and $z_2$ satisfying $|z_1| = 12$ and $|z_2 - (3 + 4i)| = 5$,the minimum value of $|z_1 - z_2|$ is:

$PQ$ and $PR$ are two infinite rays. $QAR$ is an arc. $A$ point lying in the shaded region,excluding the boundary,satisfies:

The number of complex numbers $z$ satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$ is:

If the complex number $z=x+iy$,where $i=\sqrt{-1}$,satisfies the condition $|z+1|=1$,then $z$ lies on