Let a>0 and let t be any real parameter. Define the point P(at2,2at) in the xy-plane.
a. (i) Show, by direct substitution, that P(at2,2at) satisfies the equation y2=4ax.
b. (ii) Compute the distance of P from the focus F(a,0) and the perpendicular distance of P from the directrix x=−a, and prove that these two distances are equal for every real t.
c. (iii) Explain briefly how the result in part (ii) confirms that the set of points {(at2,2at)∣t∈R} exhausts the entire parabola y2=4ax.