College Algebra 9th txtbk

Review Exercises 421
CHAPTER
6 Review Exercises
Work through all the problems in this chapter review and check
answers in the back of the book. Answers to all review problems
are there, and following each answer is a number in italics
indicating the section in which that type of problem is discussed.
Where weaknesses show up, review appropriate sections in the text.
In Problems 1–6, graph each equation and locate foci. Locate the
directrix for any parabolas. Find the lengths of major, minor,
transverse, and conjugate axes where applicable.
1. 9x 2 25y 2 225 2. x 2 12y
3. 25y 2 9x 2 225 4. x 2 y 2 16
5. y 2 8x 6. 2x 2 y 2 8
7. Find the equation of the parabola having its vertex at the origin,
its axis of symmetry the x axis, and (4, 2) on its graph.
In Problems 8 and 9, find the equation of the ellipse in the form
x 2
M, N 7 0
M y2
N 1
if the center is at the origin, and:
8. Major axis on x axis
Major axis length 12
Minor axis length 10
9. Major axis on y axis
Minor axis length 12
Distance between foci 16
In Problems 10 and 11, find the equation of the hyperbola in the
form
x 2
y 2
or M, N 0
M x2
M y2
N 1 N 1
if the center is at the origin, and:
10. Transverse axis on y axis
Conjugate axis length 6
Distance between foci 8
11. Transverse axis on x axis
Transverse axis length 14
Conjugate axis length 16
12. Find the equation of the parabola having directrix y 5 and
focus (0, 5).
13. Find the foci of the ellipse through the point (6, 0) if the center
is at the origin, the major axis is on the x axis, and the major axis
has twice the length of the minor axis.
14. Find the y intercepts of a hyperbola if the center is at the origin,
the conjugate axis is on the x axis and has length 4, and (0, 3)
is a focus.
15. Find the directrix of a parabola having its vertex at the origin and
focus (4, 0).
16. Find the points of intersection of the parabolas x 2 8y and
y 2 x.
17. Find the x intercepts of an ellipse if the center is at the origin,
the major axis is on the y axis and has length 14, and (0, 1) is
a focus.
18. Find the foci of the hyperbola through the point (0, 4) if the
center is at the origin, the transverse axis is on the y axis, and the
conjugate axis has twice the length of the transverse axis.
19. Use the definition of a parabola and the distance formula
to find the equation of a parabola with directrix x 6 and focus
at (2, 4).
20. Find an equation of the set of points in a plane each of whose
distance from (4, 0) is twice its distance from the line x 1.
Identify the geometric figure.
21. Find an equation of the set of points in a plane each of whose
distance from (4, 0) is two-thirds its distance from the line
x 9. Identify the geometric figure.
In Problems 22–24, find the equations of the asymptotes of each
hyperbola.
x 2
y 2
22. 23.
64 x2
49 y2
25 1
4 1
24. 4x 2 y 2 1
APPLICATIONS
25. COMMUNICATIONS A parabolic satellite television antenna has
a diameter of 8 feet and is 1 foot deep. How far is the focus from the
vertex?
26. ENGINEERING An elliptical gear is to have foci 8 centimeters
apart and a major axis 10 centimeters long. Letting the x axis lie
along the major axis (right positive) and the y axis lie along the minor
axis (up positive), write the equation of the ellipse in the standard
form
x 2
a 2 y2
b 2 1
27. SPACE SCIENCE A hyperbolic reflector for a radio telescope (such
as that illustrated in Problem 63, Exercises 6-3) has the equation
y 2
40 2 x2
30 2 1
If the reflector has a diameter of 30 feet, how deep is it? Compute the
answer to three significant digits.