Ellipse and complex numbers?

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

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