If $P, Q, R, S$ are represented by the complex numbers $4 + i, 1 + 6i, -4 + 3i, -1 - 2i$ respectively,then $PQRS$ is a

If $z$ is a complex number such that $\frac{z - 1}{z + 1}$ is purely imaginary,then

Let $\omega = z \bar{z} + k_1 z + k_2 i z + \lambda(1 + i)$,where $k_1, k_2 \in R$. Let $\operatorname{Re}(\omega) = 0$ be the circle $C$ of radius $1$ in the first quadrant touching the line $y = 1$ and the $y$-axis. If the curve $\operatorname{Im}(\omega) = 0$ intersects $C$ at $A$ and $B$,then $30(AB)^2$ is equal to $.......$.

Let $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2$,where $z_1, z_2 \in \mathbb{C}$. Then the minimum value of $|z_1 - z_2|$ is:

If $\frac{|3z - i|}{|4z - 2 + 3i|} = K$ $(K \in \mathbb{R}^+)$ represents a straight line,then the value of $K$ is:

If $z = (\lambda + 3) + i\sqrt{5 - \lambda^2}$,then the locus of $z$ is a