In this exercise we show that if the point P(x, y) in the accompanying figure satisfies

Chapter 11, Problem 16

(choose chapter or problem)

In this exercise we show that if the point P(x, y) in the accompanying figure satisfies the condition then, in fact, the point P(x, y) lies on the ellipse x2 /a2 y2 /b2 1. (a) From the given equation we have (F2P) 2 e2 (PD) 2 . Use the distance formula and the figure to deduce from this equation that (b) In the equation in part (a), replace e with c/a. After carrying out the indicated operations and simplifying, show that the equation can be written (c) The equation in part (b) is equivalent to (a2 c2 )x2 a2 y2 a2 (a2 c2 ). Now replace the quantity a2 c2 by b2 and show that the resulting equation can be written x2 /a2 y2 /b2 1; thus P lies on the ellipse, as we wished to show. (d) If then let to show that the conic section is the hyperbola with equation x2 a2 y2 b2 1 b2