r
• Review Exercises
In Exercises 1—6, match the equation with the
correct graph. [The graphs are labeled (a), (b), (c),
(d), (e), and (f).j
(a)
4
112
82
In Exercises 17—20, sketch the graph of the parabola
Use a graphing utility to verify your graph.
0
2
18. 8y+x
20.y_8x2=0
y
0
2
17.4x—
=
19.l,v2+18x=0
In Exercises 2 1—24, write the equation of the specifie
ellipse with center at the origin.
21. Vertices: (±5, 0); Foci: (±4, 0)
22. Vertices: (0, ±10); Foci: (0,±8)
23. Vertices: (0, ±6); Passes through: (2, 2)
24. Vertices: (±7, 0); Passes through: (0, ±6)
(c)
In Exercises 25—28, sketch the graph of the ellips
Use a graphing utility to verify your graph.
2
2 + v
25. 4x
(e)
26.
16
2
+ 6v
2
=
18
x
8y
48
28.3
+
2
=
6
+4v
2
6x
27. 3
In Exercises 29—32, write the equation of the specifie
hyperbola with center at the origin.
2 +2
1. 4x
y
2 + 4
3. x
2
y
5. 4x
2
—
2
y
=
4
2. 2
y = —4x
4 2 + )12 = 49
4
6. y
2
4
=
—
y=-2x
:‘
+
—
144
=
1
12.
+
=
I
=
-
=
13. Vertex: (0, 0);
Vertical axis
Passes through: (1, 2)
14. Vertex: (0, 0);
Passes through: (4, —2)
L
15. Vertex: (0, 0);
Focus: (—6, 0)
16. Vertex: (0, 0);
Focus: (0, 3)
6
16
In Exercises 13—16, write the equation of the specified
parabola.
Vertical axis
I
(—1, 0)’’(1 0)
2 + v
8. 16x
2
0
7. x
2 + 2Oy
49
yJ
4
2
4x
In Exercises 7—12, identify the conic.
11
30.
29.
—4
—4
31. Vertices: (O,±1);
32. Vertices: (±4, 0);
Foci: (0,±3)
Foci: (±6, 0)
In Exercises 33—36, sketch the graph of the hyperbol
Use a graphing utility to verify your graph.
33
9
64
0
35. 2
5y—4x
2
34
49
36
36.x2_y2=
In Exercises 37—44, identify the conic.
6x+2v+90
37. 2
x
—
38. y
— 12v—8x+20=0
2
y+
8
l
+9y lOx— 250
x
+
39. 2
r
• Review Exercises
In Exercises 1—6, match the equation with the
correct graph. [The graphs are labeled (a), (b), (c),
(d), (e), and (f).j
(a)
4
112
82
In Exercises 17—20, sketch the graph of the parabola
Use a graphing utility to verify your graph.
0
2
18. 8y+x
20.y_8x2=0
y
0
2
17.4x—
=
19.l,v2+18x=0
In Exercises 2 1—24, write the equation of the specifie
ellipse with center at the origin.
21. Vertices: (±5, 0); Foci: (±4, 0)
22. Vertices: (0, ±10); Foci: (0,±8)
23. Vertices: (0, ±6); Passes through: (2, 2)
24. Vertices: (±7, 0); Passes through: (0, ±6)
(c)
In Exercises 25—28, sketch the graph of the ellips
Use a graphing utility to verify your graph.
2
2 + v
25. 4x
(e)
26.
16
2
+ 6v
2
=
18
x
8y
48
28.3
+
2
=
6
+4v
2
6x
27. 3
In Exercises 29—32, write the equation of the specifie
hyperbola with center at the origin.
2 +2
1. 4x
y
2 + 4
3. x
2
y
5. 4x
2
—
2
y
=
4
2. 2
y = —4x
4 2 + )12 = 49
4
6. y
2
4
=
—
y=-2x
:‘
+
—
144
=
1
12.
+
=
I
=
-
=
13. Vertex: (0, 0);
Vertical axis
Passes through: (1, 2)
14. Vertex: (0, 0);
Passes through: (4, —2)
L
15. Vertex: (0, 0);
Focus: (—6, 0)
16. Vertex: (0, 0);
Focus: (0, 3)
6
16
In Exercises 13—16, write the equation of the specified
parabola.
Vertical axis
I
(—1, 0)’’(1 0)
2 + v
8. 16x
2
0
7. x
2 + 2Oy
49
yJ
4
2
4x
In Exercises 7—12, identify the conic.
11
30.
29.
—4
—4
31. Vertices: (O,±1);
32. Vertices: (±4, 0);
Foci: (0,±3)
Foci: (±6, 0)
In Exercises 33—36, sketch the graph of the hyperbol
Use a graphing utility to verify your graph.
33
9
64
0
35. 2
5y—4x
2
34
49
36
36.x2_y2=
In Exercises 37—44, identify the conic.
6x+2v+90
37. 2
x
—
38. y
— 12v—8x+20=0
2
y+
8
l
+9y lOx— 250
x
+
39. 2
Chapter 11
28
•
Topics in Analytic Geometry
In Exercises 59—62, sketch the curve represented by
the parametric equations.
0
2 — 1 6x + 24y — 3
2 + l6y
0. 1 6x
—11 =0
4y
1. 2
4x+8y
—
x
4
—
70
9y
18y+
.2. 2
lOx+
—
x
+
— 16x+ 15=0
±
2
3. 4x
4. 9x
2 y
2 — 72x+ 8)’ + 119
59. x
y
0
Focus: (4, 0)
Directrix: x
=
—3
18. Vertex: (2, 2);
Directrix: y
=
0
1
5t
2t + 5
—
xt
2
2
61.
+
n Exercises 45—48, find a rectangular equation of the
pecified parabola.
5. Vertex: (4, 2);
16. Vertex: (2, 0);
17. Vertex: (0, 2);
=
Focus: (0, 0)
=
60. x
=
=
4t + 1
8
3r
—
62.x=
y
2 — 3
4t
=
—
1
In Exercises 63—72, sketch the curve represented by
the parametric equations and write the correspond.
ing rectangular equation by eliminating the parame.
ter. Verify your result using a graphing utility.
63.x=/
64.x=t
y=./
n Exercises 49—52, find a rectangular equation of the
pecified ellipse.
19. Vertices: (—3, 0), (7, 0); Foci: (0, 0), (4,0)
50. Vertices: (2, 0), (2, 4); Foci: (2, 1), (2, 3)
51. Vertices: (0, 1), (4, 1);
Endpoints of the minor axis: (2, 0), (2, 2)
65.x=-
66.x=t
y=t
67. x
y
y=—
68. x
=
y
52. Vertices: (—4, —1), (—4, 11);
Endpoints of the minor axis: (—6,5), (—2,5)
69.x
y
=
[n Exercises 53—56, find a rectangular equation of the
specified hyperbola.
53. Vertices: (—10, 3), (6, 3); Foci: (— 12, 3), (8, 3)
54. Vertices: (2, 2), (—2, 2); Foci: (4, 2), (—4, 2)
55. Foci: (0, 0), (8, 0); Asymptotes:)’ = ±2(x 4)
71. x
=
3
y
=
t
t
=
t + 4
= 12
70.x==
2t + 3
72. x = I
y=2
In Exercises 73—76, find two different sets of para
metric equations for the given rectangular equation.
—
Asymptotes: y
56. Foci: (3, ±2);
±2(x — 3)
In Exercises 57 and 58, complete the table for
each set of parametric equations.
57. x=3t—2andy=7—4t
-i
oH
4
1
randy
=
74.v10—x
=2x
5x
76.y
+
3
In Exercises 77—80, plot the point in the polar
coordinate system and find three additional polar
representations of the point, using —2 < 0 < 217.
23
79.
;-
58. x
73.y6x+2
”x
2
75.y
+
2
(, )
(,-)
78. (—2,
80.
)
(/)
In Exercises 81—84, plot the point given in the polar
coordinates and find the corresponding rectangular
coordinates for the point.
81.
(s _)
82. (_3)
Chapter 11
834
•
Topics in Analytic Geometiy
distinguishable permutations of
In Exercises 25 and 26, find the number of
the group of letters.
26. A, N, T, A, R, C, T, I, C, A
25. B, A, S. K, B, T, B, A, L, L
sketch its graph.
In Exercises 27—30, identify the conic and
2)2
+ 1)2
(x
(5)2
+
ci+3)
2
28
9
27
4
121
36
0
y + 1
4
2x—
2
2 +y
30. x
2 = 16
x
29. y
2
for the graph of the conic.
In Exercises 31—33, find an equation
33.
32.
—
—
—
—
4
10
4
12
foci (0, 0) and (0, 4), and asymptotes
34. Find an equation of the hyperbola with
= ±x + 2.
sented by the parametric equa
In Exercises 35—37, (a) sketch the curve repre
your graph, and (c) eliminate the
tions, (b) use a graphing utility to verify
ngular equation whose graph
parameter and write the corresponding recta
represents the curve.
37. x4lnt
36. x = 8 + 3t
35. x = 2t + 1
I.,
V
=
polar coordinates and find three
In Exercises 38—41, plot (lie given point in
IT.
< 0 < 2
additional polar representations for —2
38. (8)
39.
(, —)
41. (—3.
40. (_2,)
42. Convert the rectangular equation
— —
8x
3y + 5
=
—-)
0 to polar form.
to rectangular form.
43. Convert the polar equation r
—
r equation.
In Exercises 44—46, graph and identify the pola
44.
1
—
45. r
3
—
2 sin 0
46. r
2 + 5 cos 0
at (14, 0) and (2, ir) and one
47. Find a polar equation of the ellipse with vertices
focus at the pole.
four different positions. in how
48. A personnel manager has 15 applicants to fill
applicants are qualified for
many ways can this be done, assuming that all the
any of the four positions?
t be arranged in the proper order
49. On a game show, the digits 3, 4, and 5 mus
ged correctly, the contes
to form the price of an appliance. If they are arran
of winning if the contestant
tant wins the appliance. What is the probability
knows that the price is at least $400?
Chapter 11
Analytic Geometry
IRUE/FALSE ITEMS
F
1.
F
2.
F
3.
F
4.
F
5.
F
6.
F
7.
F
8.
F
9.
F
10.
equals the distance from that point to the directrit
On a parabola, the distance from any point to the focus
The foci of an ellipse lie on its minor axis.
The foci of a hyperbola lie on its transverse axis.
asymptotes.
Hyperbolas always have asymptotes, and ellipses never have
A hyperbola never intersects its conjugate axis.
A hyperbola always intersects its transverse axis.
l2y 0 defines an ellipse if a > 0.
2
2 + 6y
The equation ax
2 = 10 defines a parabola if b
2 + bxy + 12y
The equation 3x
—
—12.
+ sin 6) defines a hyperbola.
If (r, 6) are polar coordinates, the equation r = 2/(2 3
Parametric equations defining a curve are unique.
REVIEW EXERCISES
for use in a Practice Test.
Blue problem numbers indicate the authors’ suggestions
la, give its vertex,
In Problems 1—20, identify each equation. If it is a parabo
center, vertices,
its
give
ola,
hyperb
a
is
it
if
give its center, vertices, and foci;
1
2
y
—16x
4.
7.
4y4
x
+
2
36
10.
2
9x + 4y
13.
16.
x
=4
4y-—4
—8x
2
y
+
16x + 18y
2
2 + 9y
4x
19.
=23
y
x
+8y
4
18x
9
+
2
—
=
—
11
and directrix; if it is an ellipse,
and asymptotes.
focus,
foci,
2.
y
2
16x
3•
5
l
+x
2
y
_
16
25
6.
8.
x
9
—
2
3y
=
9.
2)’
12.
--2
y
4y=x
2
2
—
2
2 + 9y
4x
4x
11.
2
x
14.
4=0
y
x
8x—4y+
4
+
2
+
15.
17.
32=0
x
16x+16y+
2
4
—
18.
20.
y
2x—2y=1
—
2
x
—
—
=
y
8
—
2
4x
=
lSy = 11
19=O
y
3x—16y+
4
+
2
—
16x
—
ed. Graph the equation.
in Problems 21—36, obtain an equation of the conic describ
30. Parabola; focus at (3, 6); directrix the line
21. Parabola; focus at (—2, 0); directrix the line
8
y=
x2
22.
23.
24.
Ellipse; center at (0,0); focus at (0,3); vertex at
(0,5)
Hyperbola; center at (0, 0); focus at (0, 4); ver
tex at (0, —2)
Parabola; vertex at (0, 0); directrix the line
y = —3
25. Ellipse; foci at (—3, 0) and (3, 0); vertex at
(4, 0)
26. Hyperbola; vertices at (—2,0) and (2,0); focus
at (4, 0)
27. Parabola; vertex at (2, —3); focus at (2, —4)
28. Ellipse; center at (—1,2); focus at (0,2); vertex
at (2,2)
29. Hyperbola; center at (—2, —3); focus at
(—4, —3); vertex at (—3, —3)
31.
Ellipse; foci at (—4, 2) and (—4, 8); vertex at
(—4, 10)
32.
Hyperbola; vertices at (—3, 3) and (5, 3); focus
at (7, 3)
33.
Center at (—1, 2); a = 3; c
parallel to the x-axis
34.
Center at (4, —2); a = 1; c
parallel to the y-axis
35.
line
Vertices at (0, 1) and (6, 1); asymptote the
3y + 2x 9 0
=
4; transverse axis
=
4; transverse axis
—
36.
line
Vertices at (4, 0) and (4, 4); asymptote the
y + 2x —10 0