Derive the Hyperbola Equation. F-c, 0) y x a T F2 (C,0) y = 1 b P(x, y) To obtain the simplest equation for an ellipse we place the foci on the x-axis at F (-c, 0) and F2 (c, 0) so the origin is halfway between them. X For later convenience, let the difference of the distance from a point on the hyperbola to the foci be 2a. Then Id (F, P)-d(P, F)| = 2a. Now use the distance formula, plug in the points to find the distance between F and Pand distance between F and P and set the difference equal to 2a.