In this exercise we show that the focal radii of the ellipse x2 /a2 y2 /b2 1 are F1P a

Chapter 11, Problem 15

(choose chapter or problem)

In this exercise we show that the focal radii of the ellipse x2 /a2 y2 /b2 1 are F1P a ex and F2P a ex. The method used here, which avoids the use of radicals, appears in the eighteenth-century text Trait analytique des sections coniques (Paris: 1707) by the Marquis de lHpital (16611704). (For another method, one that does use radicals, see Exercise 55 in Section 11.4.) For convenience, let d1 F1P and d2 F2P, as indicated in the accompanying figure. ((a) Using the distance formula, verify thatd21 1x c22 y2 and d22 1x c22 y2y (b) Use the two equations in part (a) to show that 4cx.(c) Explain why d1 d2 2a.(d) Factor the left-hand side of the equation in part (b)and substitute for one of the factors using the equationin part (c). Show that the result can be writtend1 d2 2cx/a.(e) Add the equations in parts (c) and (d). Show that theresulting equation can be written d1 a ex, asrequired.(f) Use the equation in part (c) and the result in part (e)to show that d2 a ex.1