Let $\displaystyle z = a + ib$ and $\displaystyle p_{k}(k = 1,2,3,.....,n)$ are the $\displaystyle n$th roots of unity. If $\displaystyle A_{k} = \Re(z)\Re(p_{k}) + \Im(z)\Im(p_{k})$, then show that $\displaystyle A_{k}$ lies on an ellipse. If $\displaystyle S$ is the focus of the ellipse on the positive major axis, then show that:

(i)$\displaystyle \sum_{k = 1}^{n} |A_{k} - S| = na$
(ii)$\displaystyle \sum_{k = 1}^{n} |A_{k} - S|^2 = \frac{n(3a^2 - b^2)}{2}$