James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

682 Chapter 10 Parametric Equations and Polar Coordinates
PF 1, PF 2 and the ellipse as shown in the figure. Prove that
− . This explains how whispering galleries and litho tripsy
work. Sound coming from one focus is reflected and passes
through the other focus. [Hint: Use the formula in Problem 21
on page 273 to show that tan − tan .]
66. Let Psx 1, y 1d be a point on the hyperbola x 2 ya 2 2 y 2 yb 2 − 1
with foci F 1 and F 2 and let and be the angles between
the lines PF 1, PF 2 and the hyperbola as shown in the figure.
Prove that − . (This is the reflection property of the hyperbola.
It shows that light aimed at a focus F 2 of a hyperbolic
mirror is reflected toward the other focus F 1.)
y
P
y
å
P(⁄, ›)
∫
F¡
å
∫
0 F x
F¡ 0 F x
≈ ¥
+ =1
a@ b@
P
F¡
F
In the preceding section we defined the parabola in terms of a focus and directrix, but
we defined the ellipse and hyperbola in terms of two foci. In this section we give a more
unified treatment of all three types of conic sections in terms of a focus and directrix.
Further more, if we place the focus at the origin, then a conic section has a simple polar
equation, which provides a convenient description of the motion of planets, satellites,
and comets.
1 Theorem Let F be a fixed point (called the focus) and l be a fixed line
(called the directrix) in a plane. Let e be a fixed positive number (called the
eccentricity). The set of all points P in the plane such that
| PF |
| Pl |
(that is, the ratio of the distance from F to the distance from l is the constant e)
is a conic section. The conic is
− e
(a) an ellipse if e , 1
(b) a parabola if e − 1
(c) a hyperbola if e . 1
Proof Notice that if the eccentricity is e − 1, then | PF | − | Pl | and so the given
condition simply becomes the definition of a parabola as given in Section 10.5.
Let us place the focus F at the origin and the directrix parallel to the y-axis and
d units to the right. Thus the directrix has equation x − d and is perpendicular to the
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