If the least and the largest real values of $\alpha,$ for which the equation $z+\alpha|z-1|+2i=0$ ($z \in \mathbb{C}$ and $i=\sqrt{-1}$) has a solution,are $p$ and $q$ respectively; then $4(p^2+q^2)$ is equal to ..........

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 $z$ be a complex number such that the real part of $\frac{z-2i}{z+2i}$ is zero. Then,the maximum value of $|z-(6+8i)|$ is equal to:

If $|z_1 + z_2| = |z_1 - z_2|$,then the difference in the amplitudes of $z_1$ and $z_2$ is

If $|z - 2|/|z - 3| = 2$ represents a circle,then its radius is equal to

$\sinh(ix)$ is equal to