Chapter 11 Review Exercise A B C

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Additional Topics in Analytic Geometry

Chapter 11 Review Exercise 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.

Obtain an equation in x and y by eliminating the parameter, and identify the curve. In Problems 11–13, graph each system of equations in the same coordinate system and find the coordinates of any points of intersection. 11. x2  4y2  32 x  2y  0

A In Problems 1–3, 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. 9x2  25y2  225

12. 16x2  25y2  400 16x2  45y  0 13.

x2  y2  10 16x2  y2  25

2. x2  12y

In Problems 14–16, transform each equation into one of the standard forms in Table 1 in the review. Identify the curve and graph it.

3. 25y2  9x2  225

14. 16x2  4y2  96x  16y  96  0

In Problems 4–6: (A) Write each equation in one of the standard forms listed in Table 1 of the review. (B) Identify the curve. 4. 4( y  2)  25(x  4)  100 2

2

5. (x  5)2  12( y  4)  0 6. 16(x  6)2  9( y  4)2  144

B 7. Find the equation of the parabola having its vertex at the origin, its axis the x axis, and (4, 27) on its graph. 8. Find an equation of an ellipse in the form x 2 y2  1 M N

M, N  0

if the center is at the origin, the major axis is on the y axis, the minor axis has length 4, and the distance between the foci is 62. 9. Find an equation of a hyperbola in the form x2 y 2  1 M N

M, N  0

if the center is at the origin, the transverse axis has length 10, and the foci are 6 units from the center. 10. Plot the curve given parametrically by

15. x2  4x  8y  20  0 16. 4x2  9y2  24x  36y  36  0 17. Given the parametric equations of a plane curve, x  2  2 sin  and y  3  4 cos , obtain an equation in x and y by eliminating the parameter. Use the simpler of the two forms to plot the curve. Identify the curve. 18. Use a graphing utility to graph x2  y and x2  50y in the viewing window 10  x, y  10. Find m so that the graph of x2  y in the viewing window m  x, y  m, has the same appearance as the graph of x2  50y in 10  x, y  10. Explain.

C 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 coordinates of any foci relative to the original coordinate system. 22. Problem 14

23. Problem 15

24. Problem 16

25. Given the parametric equations of a plane curve

x  t 2

x  2t

y  12 t 2  1

y  2t

Cumulative Review Exercise Chapters 10 and 11

obtain an equation in x and y by eliminating the parameter. Use the simpler of the two forms to graph the curve. Identify the curve.

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

26. Use a graphing utility to find, to two decimal places, the coordinates of all points of intersection of x2  3y2  9x  7y  22  0 and 4x2  5x  10y  53  0.

APPLICATIONS

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x2 y2  1 a2 b2 29. Space Science. A hyperbolic reflector for a radiotelescope (such as that illustrated in Problem 41, Exercise 11-3) has the equation

27. 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?

y2 x2  21 2 40 30

28. Engineering. An elliptical gear is to have foci 8 centimeters apart and a major axis 10 centimeters long. Letting the

If the reflector has a diameter of 30 feet, how deep is it? Compute the answer to 3 significant digits.

Cumulative Review Exercise Chapters 10 and 11 Work through all the problems in this cumulative 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 7–9, graph each equation and locate foci. Locate the directrix for any parabolas. Find the lengths of major, minor, transverse, and conjugate axes where applicable. 7. 25x2  36y2  900 8. 25x2  36y2  900 9. 25x2  36y  0

A 1. Determine whether each of the following can be the first three terms of an arithmetic sequence, a geometric sequence, both, or neither. (A) 5, 25, 100, . . . (B) 15, 3, 9, . . . (C) 1, 1, 1, . . . (D) 64, 16, 4, . . . (E) 17, 119, 833, . . . (F) 1, 3, 6, . . . In Problems 2–4: (A) Write the first four terms of each sequence. (B) Find a8. (C) Find S8. 2. an  (2)n

10. A coin is flipped three times. How many combined outcomes are possible? Solve: (A) By using a tree diagram (B) By using the multiplication principle 11. How many ways can 4 distinct books be arranged on a shelf? Solve: (A) By using the multiplication principle (B) By using permutations or combinations, whichever is applicable 12. Plot the curve given parametrically by

3. an  6n  5

x  2t  3

4. a1  20; an  an1  4, n 2

y  4t  5

5. Evaluate each of the following: 10! 25! (A) 7! (B) (C) 22! (10  4)! 4! 6. Evaluate each of the following: 12 (A) (B) C9,4 (C) P8,5 6

 

Obtain an equation in x and y by eliminating the parameter, and identify the curve. Verify Problems 13 and 14 for n  1, 2, and 3. 13. Pn: 1  5  9  . . .  (4n  3)  n(2n  1) 14. Pn: n2  n  2 is divisible by 2