If $z = x + iy$ and $z^2 = (i \bar{z})^2$,then

Let $z=x+iy$ represent a point $P(x, y)$ in the Argand plane. If $z$ satisfies the condition that $\text{arg}\left(\frac{z-3}{z-2i}\right)=-\frac{\pi}{2}$,then the locus of $P$ is

Suppose $z \in \mathbb{C}$ has an argument $\theta$ such that $0 < \theta < \frac{\pi}{2}$ and satisfies the equation $|z - 3i| = 3$. What is the value of $\cot \theta - \frac{6}{z}$?

If $\cos \alpha + 2 \cos \beta + 3 \cos \gamma = 0$,$\sin \alpha + 2 \sin \beta + 3 \sin \gamma = 0$ and $\alpha + \beta + \gamma = \pi$,then $\sin 3 \alpha + 8 \sin 3 \beta + 27 \sin 3 \gamma$ is equal to

The locus of a point $z$ satisfying $|z|^2 = \operatorname{Re}(z)$ is a circle with centre

Let $z$ be the complex number satisfying $|z-5| \le 3$ and having the maximum positive principal argument. Then $34|\frac{5z-12}{5iz+16}|^{2}$ is equal to: