When $\frac{z + i}{z + 2}$ is purely imaginary,the locus described by the point $z$ in the Argand diagram is a

$POQ$ is a straight line through the origin $O$. $P$ and $Q$ represent the complex numbers $z_1 = a + ib$ and $z_2 = c + id$ respectively. If $OP = OQ$,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 $.......$.

If $z = x + iy$ and $|z - zi| = 1,$ then

If $a = \operatorname{Im}\left(\frac{1+z^2}{2iz}\right)$ and $z$ is any non-zero complex number such that $|z|=1$,then $a=$

$z_1$ and $z_2$ are the roots of $3z^2 + 3z + b = 0$. If the origin,$A(z_1)$,and $B(z_2)$ form an equilateral triangle,then the value of $b$ is equal to