$z=x+i y$ and the point $P$ represents $z$ in the argand plane. If the amplitude of $\left(\frac{2 z-i}{z+2 i}\right)$ is $\frac{\pi}{4}$, then the equation of the locus of $P$ is

$\alpha, \beta$ are the roots of the equation $x^{2}+2 x+4=0$. If the point representing $\alpha$ in the argand diagram lies in the 2nd quadrant and $\alpha^{2024}-\beta^{2024}=i k,(i=\sqrt{-1})$, then $k=$

If $z=x+i y$ satisfies the equation $z^{2}+a z+a^{2}=0, a \in R$, then

If $z_{1}, z_{2}, z_{3}$ are three complex numbers with unit modulus such that $\left|z_{1}-z_{2}\right|^{2}+\left|z_{1}-z_{3}\right|^{2}=4$, then $z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}+z_{1} \bar{z}_{3}+\bar{z}_{1} z_{3}=$