Answer the following questions regarding the constant–sum property of an ellipse whose centre is the origin and whose major axis lies on the x–axis.
(i) Write the relation connecting the semi-major axis a, the semi-minor axis b, and the focal distance c of such an ellipse.
(ii) Let P(x,y) be any point satisfying a2x2+b2y2=1.
Show, without eliminating the square roots explicitly, that the sum of its distances from the two foci F1(−c,0) and F2(c,0) is the constant 2a. Give a clear justification for every step of your reasoning.