If $z_{1}=\sqrt{3}+i \sqrt{3}$ and $z_{2}=\sqrt{3}+i$, and $\left(\frac{z_{1}}{z_{2}}\right)^{50}=x+i y$, then the point $(x, y)$ lies in

The roots of the equation $x^{3}-3 x^{2}+3 x+7=0$ are $\alpha, \beta, \lambda$ and $\omega, \omega^{2}$ are complex cube roots of unity, If the terms containing $x^{2}$ and $x$ are missing in the transformed equation when each one of these roots is decreased by $h$, then $\frac{\alpha-h}{\beta-h}+\frac{\beta-h}{\gamma-h}+\frac{\gamma-h}{\alpha-h}=$

If $x$ and $y$ are two positive real numbers such that $x+i y=\frac{13 \sqrt{-5+12 i}}{(2-3 i)(3+2 i)}$, then $13 y-26 x=$

If $z=x+i y$ and if the point $P$ represents $z$ in the argand plane, then the locus of $z$ satisfying the equation $|z-1|+|z+i|=2$ is