Let z = a + ib and p_{k}(k = 1,2,3,.....,n) are the nth 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}