For $a, b, c, d \in R$, if $z_1 = a + ib$ and $z_2 = c + id$ are such that $|z_1| = |z_2| = 1$ and $\operatorname{Re}(z_1 \bar{z}_2) = 0$, then the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfy

Suppose that $z_{1}, z_{2}, z_{3}$ are three vertices of an equilateral triangle in the Argand plane. Let $\alpha = \frac{1}{2}(\sqrt{3} + i)$ and $\beta$ be a non-zero complex number. The points $\alpha z_{1} + \beta, \alpha z_{2} + \beta, \alpha z_{3} + \beta$ will be

$A$ 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:

If $\frac{z-1}{2z+1}$ is a purely imaginary number,then the locus of $z$ represents a circle. Find its radius.

The points $z_1, z_2, z_3, z_4$ in the complex plane are the vertices of a parallelogram taken in order,if and only if

Let $z=x+iy$ be a complex number with $x, y \in \mathbb{Z}$. Then,the area (in sq units) of the rectangle whose vertices are the roots of the equation $\bar{z} \cdot z^3+z \cdot \bar{z}^3=350$ is