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

678 Chapter 10 Parametric Equations and Polar Coordinates
significant application of hyperbolas was found in the navigation systems developed in
World Wars I and II (see Exercise 51).
Notice that the definition of a hyperbola is similar to that of an ellipse; the only change
is that the sum of distances has become a difference of distances. In fact, the derivation of
the equation of a hyperbola is also similar to the one given earlier for an ellipse. It is left
as Exercise 52 to show that when the foci are on the x-axis at s6c, 0d and the difference
of dis tances is | PF 1 | 2 | PF 2 | − 62a, then the equation of the hyperbola is
6
x 2
a 2 2 y 2
b 2 − 1
(_a, 0)
b
y=_ x
a
(_c, 0)
FIGURE 12
x 2
a 2 2 y 2
y
b
y= x
a
(a, 0)
0 (c, 0) x
where c 2 − a 2 1 b 2 . Notice that the x-intercepts are again 6a and the points sa, 0d and
s2a, 0d are the vertices of the hyperbola. But if we put x − 0 in Equation 6 we get
y 2 − 2b 2 , which is impossible, so there is no y-intercept. The hyperbola is symmetric
with respect to both axes.
To analyze the hyperbola further, we look at Equation 6 and obtain
x 2
a − 1 1 y 2
2 b > 1 2
This shows that x 2 > a 2 , so | x | − sx 2 > a. Therefore we have x > a or x < 2a. This
means that the hyperbola consists of two parts, called its branches.
When we draw a hyperbola it is useful to first draw its asymptotes, which are the
dashed lines y − sbyadx and y − 2sbyadx shown in Figure 12. Both branches of the
hyperbola approach the asymptotes; that is, they come arbitrarily close to the asymptotes.
(See Exercise 4.5.73, where these lines are shown to be slant asymptotes.)
7 The hyperbola
x 2
a 2 2 y 2
b 2 − 1
has foci s6c, 0d, where c 2 − a 2 1 b 2 , vertices s6a, 0d, and asymptotes
y − 6sbyadx.
y
(0, c)
If the foci of a hyperbola are on the y-axis, then by reversing the roles of x and y we
obtain the following information, which is illustrated in Figure 13.
b 2 − 1 0
a
y=_ b x
a
y= b x
(0, a)
(0, _a)
x
8 The hyperbola
y 2
a 2 x 2
2 b − 1 2
has foci s0, 6cd, where c 2 − a 2 1 b 2 , vertices s0, 6ad, and asymptotes
y − 6saybdx.
(0, _c)
FIGURE 13
y 2
a 2 x 2
2 b − 1 2
Example 4 Find the foci and asymptotes of the hyperbola 9x 2 2 16y 2 − 144 and
sketch its graph.
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