If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$, then $k=$

Let $z=x+i y$ be a point in the argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $z$ is

If a point $P$ denotes the complex number $z=x+i y$ in the argand plane and if $\frac{z-(2+i)}{z+(1-2 i)}$ is purely real, then the locus of $P$ is

If $i$ is the root of the equation $x^2+1=0$, then

$$ (1+\sqrt{3} i)^{2023}+(1-\sqrt{3} i)^{2023}= $$