If the imaginary part of $\frac{2 z+1}{i z+1}$ is $-2$,then the locus of the point representing $z$ in the complex plane is

If $a$ and $c$ are complex numbers and $b$ is a real number in the Argand plane,then the perpendicular distance from $c$ to the line $a \bar{z} + \bar{a} z + b = 0$ is

If $a, b, c$ and $u, v, w$ are complex numbers representing the vertices of two triangles such that $c = (1 - r)a + rb$ and $w = (1 - r)u + rv$,where $r$ is a complex number,then the two triangles

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

If ${z_1} = 10 + 6i$,${z_2} = 4 + 6i$ and $z$ is a complex number such that $\text{amp}\left( \frac{z - z_1}{z - z_2} \right) = \frac{\pi}{4}$,then the value of $|z - 7 - 9i|$ is equal to

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 $.......$.