College Algebra 9th txtbk

412 CHAPTER 6 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
MATCHED PROBLEM 4
Find an equation of a hyperbola in the form
x 2
M y2
N 1
M, N 7 0
if the center is at the origin, and:
(A) Length of transverse axis is 50 (B) Length of conjugate axis is 12
Length of conjugate axis is 30 Distance of foci from center is 9
ZZZ EXPLORE-DISCUSS 2
(A) Does the line with equation y x intersect the hyperbola with equation
x 2 (y 2 4) 1? If so, find the coordinates of all intersection points.
(B) Does the line with equation y 3x intersect the hyperbola with equation
x 2 ( y 2 4) 1? If so, find the coordinates of all intersection points.
(C) For which values of m does the line with equation y mx intersect the hyperbola
? Find the coordinates of all intersection points.
x 2
a y2
2 b 1 2
Z Applications
You may not be aware of the many important uses of hyperbolic forms. They are encountered
in the study of comets; the loran system of navigation for pleasure boats, ships, and
aircraft; sundials; capillary action; nuclear reactor cooling towers; optical and radio telescopes;
and contemporary architectural structures. The TWA building at Kennedy Airport
is a hyperbolic paraboloid, and the St. Louis Science Center Planetarium is a hyperboloid.
With such structures, thin concrete shells can span large spaces [Fig. 8(a)]. Some comets
from outer space occasionally enter the sun’s gravitational field, follow a hyperbolic path
around the sun (with the sun as a focus), and then leave, never to be seen again [Fig. 8(b)].
Example 5 illustrates the use of hyperbolas in navigation.
Z Figure 8 Uses of hyperbolic
forms.
Comet
Sun
St. Louis Planetarium
(a)
Comet around sun
(b)
EXAMPLE 5 Navigation
A ship is traveling on a course parallel to and 60 miles from a straight shoreline. Two transmitting
stations, S 1 and S 2 , are located 200 miles apart on the shoreline (Fig. 9). By timing
radio signals from the stations, the ship’s navigator determines that the ship is between
the two stations and 50 miles closer to S 2 than to S 1 . Find the distance from the ship to
each station. Round answers to one decimal place.