College Algebra 9th txtbk

SECTION 6–3 Hyperbola 417
54. Consider the hyperbola with equation
ECCENTRICITY Problems 59 and 60 (and Problems 45 and
46 in Exercises 6-2) are related to a property of conics called
y 2
a x2
2 b 1 eccentricity, which is denoted by a positive real number E.
2 Parabolas, ellipses, and hyperbolas all can be defined in terms of E,
a fixed point called a focus, and a fixed line not containing the
(A) Show that y a b2
b x 21 x 2.
focus called a directrix as follows: The set of points in a plane
(B) Explain why the hyperbola approaches the lines y a each of whose distance from a fixed point is E times its distance
b x
from a fixed line is an ellipse if 0 E 1, a parabola if E 1,
as |x| becomes larger.
and a hyperbola if E 1.
(C) Does the hyperbola approach its asymptotes from above or
below? Explain.
59. Find an equation of the set of points in a plane each of whose
55. Let F and F be two points in the plane and let c be a constant
such that c d(F, F). Describe the set of all points P in the
plane such that the absolute value of the difference of the distances
from P to F and F is equal to the constant c.
56. Let F and F be two points in the plane and let c denote the constant
d(F, F). Describe the set of all points P in the plane such
that the absolute value of the difference of the distances from P
to F and F is equal to the constant c.
57. Study the following derivation of the standard equation
of a hyperbola with foci (c, 0), x intercepts (a, 0), and endpoints
of the conjugate axis (0, b). Explain why each equation
follows from the equation that precedes it. [Hint: Recall
that c 2 a 2 b 2 .]
|d 1 d 2 | 2a
2(x c) 2 y 2 2a 2(x c) 2 y 2
(x c) 2 y 2 4a 2 4a2(x c) 2 y 2 (x c) 2 y 2
2(x c) 2 y 2 a cx
a
(x c) 2 y 2 a 2 2cx c2 x 2
a 2
distance from (3, 0) is three-halves its distance from the line
x 4 3. Identify the geometric figure.
60. Find an equation of the set of points in a plane each of whose
distance from (0, 4) is four-thirds its distance from the line
y 9 4. Identify the geometric figure.
APPLICATIONS
61. ARCHITECTURE An architect is interested in designing a thinshelled
dome in the shape of a hyperbolic paraboloid, as shown in
Figure (a). Find the equation of the hyperbola located in a coordinate
system [Fig. (b)] satisfying the indicated conditions. How far
is the hyperbola above the vertex 6 feet to the right of the vertex?
Compute the answer to two decimal places.
Hyperbola
a1 c2
a 2b x2 y 2 a 2 c 2
x 2
a 2 y2
b 2 1
58. Study the following derivation of the standard equation
of a hyperbola with foci (0, c), y intercepts (0, a), and endpoints
of the conjugate axis (b, 0). Explain why each equation
follows from the equation that precedes it. [Hint: Recall
that c 2 a 2 b 2 .]
|d 1 d 2 | 2a
2x 2 ( y c) 2 2a 2x 2 ( y c) 2
x 2 ( y c) 2 4a 2 4a2x 2 ( y c) 2 x 2 ( y c) 2
2x 2 ( y c) 2 a cy a
x 2 (y c) 2 a 2 2cy c2 y 2
x 2 a1 c2
a 2b y2 a 2 c 2
y 2
a 2 x2
b 2 1
a 2
Parabola
Hyperbolic paraboloid
(a)
10
10
y
Hyperbola part of dome
(b)
62. NUCLEAR POWER A nuclear reactor cooling tower is a
hyperboloid, that is, a hyperbola rotated around its conjugate axis,
as shown in Figure (a) on page 418. The equation of the hyperbola
in Figure (b) used to generate the hyperboloid is
x 2
100 2 y2
(8, 12)
x
10
150 1 2