If $z=x+iy, x, y \in R$ and the imaginary part of $\frac{\bar{z}-1}{\bar{z}-i}$ is $1$,then the locus of $z$ is

For any two non-zero complex numbers $z_1$ and $z_2$, if $|z_1+z_2|^2=|z_1|^2+|z_2|^2$, then

If ${z_1}, {z_2}, {z_3}, {z_4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|$,then ${z_1}, {z_2}, {z_3}, {z_4}$ are

$A$ function $f$ is defined on the complex numbers by $f(z) = (a + ib)z$,where $a, b \in \mathbb{R}^+$. This function has the property that the $f$-image of any point in the complex plane is equidistant from that point and the origin. If $|a + bi| = 10$ and $b^2 = \frac{p}{q}$,where $p, q \in \mathbb{Z}$ and $\text{gcd}(p, q) = 1$,then $p + q$ is:

The inequality $|z - 4| < |z - 2|$ represents the region given by

If $z$ is a complex number such that $|z| \ge 2$,then the minimum value of $|z + \frac{1}{2}|$ is: