If $\alpha$ and $\beta$ are the complex cube root of unity, show that ${\alpha }^2+{\beta }^2+\alpha \beta =0$.

If $\alpha$ and $\beta$ are the complex cube roots of unity, show that ${\alpha }^4+{\beta }^4+{\alpha }^{-1}{\beta }^{-1}=0$.

Find the equation in cartesian coordinates of the locus of $z$ if $\left|z\right| =10$.

Find the equation in cartesian coordinates of the locus of $z$ if $|z-3|=2$.

Find the equation in cartesian coordinates of the locus of $z$, if $|z-5+6 i|=5$.

Find the equation in cartesian coordinates of the locus of $z$, if $|z+8|=|z-4|$.

Find the equation in cartesian coordinates of the locus of $z$, if $|z-2-2i|=|z+2+2i|$.

Find the equation in cartesian coordinates of the locus of $z$ if

$\frac{|z+3i|}{|z-6i|}=1$