In Section 11.2, Example 6,we saw that the center of an
ellipse may be shifted away from the origin. The same process
applies to hyperbolas. For example, the hyperbola shown at
the right,
has the same graph as
but it is centered at. Graph each hyperbola with center shifted away from
the origin.
31. 32.
33. 34.
Solve each problem.
35.Two buildings in a sports complex are shaped and po-
sitioned like a portion of the branches of the hyperbola
with equation
where xand yare in meters.
(a)How far apart are the buildings at their closest
point?
(b) Find the distance din the figure.
36.In rugby, after a try(similar to a touch-
down in American football) the scoring
team attempts a kick for extra points. The
ball must be kicked from directly behind
the point where the try was scored. The
kicker can choose the distance but cannot
move the ball sideways. It can be shown
that the kicker’s best choice is on the
hyperbola with equation
where 2gis the distance between the goal
posts. Since the hyperbola approaches its
asymptotes, it is easier for the kicker to
estimate points on the asymptotes instead
of on the hyperbola. What are the asymptotes of this hyperbola? Why is it relatively easy
to estimate them? (Source:Isaksen, Daniel C., “How to Kick a Field Goal,” The College
Mathematics Journal.)
x^2
g^2
-
y^2
g^2
=1,
400 x^2 - 625 y^2 =250,000,
1 y- 522
9
-
x^2
25
= 1
y^2
36
-
1 x- 222
49
= 1
1 x+ 322
16
-
1 y- 222
25
= 1
1 x- 222
4
-
1 y+ 122
9
= 1
1 - 5, 2 2
x^2
4
-
y^2
9
=1,
1 x+ 522
4
-
1 y- 222
9
=1,
656 CHAPTER 11 Nonlinear Functions, Conic Sections, and Nonlinear Systems
x
y
Shift 2
units up
(0, 0)
Shift 5
units left
(–5, 2 )
d
50 m
NOT TO SCALE
y
x
