Chapter 13
Conic Sections and Systems of Nonlinear Equations
13.1 Check Points
1.
22
2
xh yk r
3.
22
22
2
314
(3) 1 2
xy
xy
The center is
3,1 and the radius is 2 units.
4.
22
22
4410
4 4 1
xy xy
xx yy
Complete the squares.
22
2
424
22
b
13.1 Concept and Vocabulary Check
1. circle; center; radius
13.1 Exercise Set
1.
22
2
22
2
22
007
xh yk r
xy
22
2
22
325
3225
xy
xy
4.
2
22
22
214
2116
xy
xy
Chapter 13 Conic Sections and Systems of Nonlinear Equations
8.
2
22
22
535
535
xy
xy
11.
22
22
2
16
004
xy
xy
The center is
0, 0
and the radius is 4 units.
12.
22
22
2
49
007
xy
xy
The center is
0, 0
and the radius is 7 units.
13.
22
22
2
3136
316
xy
xy
14.
22
22
2
2316
234
xy
xy
15.
22
222
324
322
xy
xy
The center is
3, 2
and the radius is 2 units.
16.
22
222
1425
145
xy
xy
The center is
1, 4
and the radius is 5 units.
17.
22
22
2
224
222
xy
xy
Section 13.1 The Circle
18.
22
4536
xy
19.
22
22
6260
6 2 6
xy xy
xx yy
Complete the squares.
The center is
3, 1
and the radius is 2 units.
20.
22
84160
xy xy
The center is
4, 2
and the radius is 2 units.
22
10 6 30
xx yy
Complete the squares.
22
2
10 525
22
b
22
2
639
b
Chapter 13 Conic Sections and Systems of Nonlinear Equations
22.
22
22
412 90
4 12 9
xy x y
xx y y
Complete the squares.
22
2
424
22
b
23.
22
22
8280
8 2 8
xy xy
xx y y
Complete the squares.
The center is
4,1
and the radius is 5 units.
24.
22
22
12 6 4 0
12 6 4
xy xy
xx yy
Complete the squares.
22
2
12 636
22
b
25.
22
22
2150
2 15
xxy
xx y
Complete the square.
Section 13.1 The Circle
26.
22
22
670
6 7
xy y
xy y
The center is
0, 3
and the radius is 4 units.
27.
Check
0, 4
:
2
2
0416
16 16 true
044
44 true
28.
Check
0, 3:
2
2
039
99 true
033
33 true
The solution set is
0, 3 , 3, 0.
29.
Check
0, 3:
22
22
02 33 9
204
303
33 true
Chapter 13 Conic Sections and Systems of Nonlinear Equations
30.
Check
3, 2
:
22
33 21 9
231
31. From the graph we can see that the center of the
circle is at
2, 1
and the radius is 2 units.
32. From the graph we can see that the center of the
circle is at
3, 1
and the radius is 3 units.
33. From the graph we can see that the center of the
circle is at
3, 2
and the radius is 1 unit.
Therefore, the equation is
34. From the graph we can see that the center of the
circle is at
1,1
and the radius is 4 units.
Therefore, the equation is
222
114
xy
b. The radius is the distance from the center to one
of the points on the circle. Using the point
3, 9 , we get:
The radius is 5 units.
c.
2
22
5105
xy
1212
,
22
xxyy
M
The center is
4,5 .
b. The radius is the distance from the center to one
of the points on the circle. Using the point
Section 13.1 The Circle
37. If we place L.A. at the origin, then we want the
equation of a circle with center at
2.4, 2.7 and
radius 30.
38. The center of the circle is (0, 68 +14) or (0, 82).
The radius of the Ferris wheel is 68 feet. The
equation of the circular wheel is:
44.
22
22
2
25
25
25
xy
yx
yx
2
136 3
yx
2
10 525
22
b
22
2
424
22
b
47. does not make sense; Explanations will vary.
Sample explanation: The radius of a circle cannot
be 0.
Chapter 13 Conic Sections and Systems of Nonlinear Equations
50. makes sense
53. false; Changes to make the statement true will vary.
A sample change is: Because the variables are not
squared, this is a linear equation, not a circle.
54. false; Changes to make the statement true will vary.
A sample change is: Because of the negative on the
constant 36, this is not a circle.
To find the area of the doughnut -shaped region, we
find the area of the larger circle and subtract the
area of the smaller circle.
dLS
AAA
56. The center of the circle with equation,
22
25
xy
, is the point (0,0). First, find the
Since the tangent line is perpendicular to the line
4
3
43
4
yx
57.
2
2
34 34 2
924162
fgx f x x
xx
3
4
x
3
52 15
515
x
Section 13.2 The Ellipse
60.
22
1
94
xy
61.
22
22
1
94
xy
62.
22
22
22
25 16 400
25 16 400
400 400 400
1
16 25
xy
xy
xy
13.2 Check Points
1.
22
1
36 9
xy
2.
22
22
16 9 144
16 9 144
xy
xy
the vertices are
0, 4 and 0, 4 .
Since
3b
and endpoints of the minor axis are
3,0 and 3,0 .
3.
22
12
1
94
xy
The center of the ellipse is
1, 2 .
Because the
denominator of the
2
termx
is greater than the
denominator of the
2
term,y
the major axis is
horizontal. Since
2
9,a
3a
and the vertices lie
3 units to the right and left of the center. Since
2
4,b
2b
and endpoints of the minor axis lie 2
units above and below the center.
Chapter 13 Conic Sections and Systems of Nonlinear Equations
4. Using the equation
22
22
1
xy
ab
the archway can
Substitute 6 for x to find the height y.
22
2
61
400 100
36 1
y
y
13.2 Concept and Vocabulary Check
1. ellipse; foci; center
2. 25;
5
; 5; (5,0); (5,0) ; 9;
3
; 3; (0, 3);
(0,3)
13.2 Exercise Set
2b
and endpoints of the minor axis are
0, 2
and 0,2 .
of the minor axis are
0, 4 and 0, 4 .
Section 13.2 The Ellipse
3.
22
1
936
xy
Because the denominator of the
2
termy
is
4.
22
1
16 49
xy
The vertices are
0, 7 and 0, 7 .
The endpoints
of the minor axis are
4, 0 and
4,0 .
5.
22
1
25 64
xy
Because the denominator of the
2
termy
is
49 36
The vertices are
7,0 and 7,0 .
The endpoints
of the minor axis are
0, 6 and
0, 6 .
7.
22
1
49 81
xy
Chapter 13 Conic Sections and Systems of Nonlinear Equations
8.
22
1
64 100
xy
The vertices are
0, 10 and 0,10 .
The
endpoints of the minor axis are
8, 0 and
8, 0 .
9.
22
22
22
25 4 100
25 4 100
100 100 100
1
425
xy
xy
xy
Because the denominator of the
2
termy
is
greater than the denominator of the
2
term,x
the
10.
22
22
22
94 36
94 36
36 36 36
1
49
xy
xy
xy
11.
22
22
22
416 64
416 64
64 64 64
1
16 4
xy
xy
xy
Section 13.2 The Ellipse
12.
22
22
16 9 144
16 9 144
144 144 144
xy
xy
13.
22
22
22
25 9 225
25 9 225
225 225 225
1
925
xy
xy
xy
Because the denominator of the
2
termy is
greater than the denominator of the
2
term,x the
14.
22
22
425100
4 25 100
100 100 100
xy
xy
15.
22
22
22
28
28
888
1
84
xy
xy
xy
Because the denominator of the
2
termx is
greater than the denominator of the
2
term,y the
Chapter 13 Conic Sections and Systems of Nonlinear Equations
16.
22
22
22
12 4 36
12 4 36
36 36 36
1
39
xy
xy
xy
The vertices are
17. From the graph, we see that the center of the ellipse
is the origin, the major axis is horizontal with a = 2,
and b = 1.
18. The center of the ellipse is the origin, and the major
axis is horizontal with a = 4, and b = 2.
19. From the graph, we see that the center of the ellipse
is the origin, the major axis is vertical with a = 2,
and b = 1.
20. The center of the ellipse is the origin, and the major
axis is vertical with a = 4, and b = 2.
22 22
22
1 or 1
416
24
xy xy
22
21
xy
5,1
2,3
The center of the ellipse is
1, 2 .
The vertices lie
4 units to the left and right of the center. The
endpoints of the minor axis lie 3 units above and
below the center.
Section 13.2 The Ellipse
23.
22
34 216
xy
The center of the ellipse is
3, 2 .
Because the
2
Center Vertices Endpoints of
Minor Axis
3, 2
34,2
7, 2
3, 2 2
3, 0
24.
22
22
39 2 36
392 36
xy
xy
The center of the ellipse is
3, 2 .
The vertices lie
6a
units to the left and right of the center. The
endpoints of the minor axis lie
2b
units above
and below the center.
Center Vertices
Endpoints of
Minor Axis
3, 2
36,2
3, 2
3, 2 2
3, 4
36,2
3, 2 2
The center of the ellipse is
4, 2 .
Because the
denominator of the
2
termy
is greater than the
denominator of the
2
term,x
the major axis is
vertical. Since
2
25,a
5a
and the vertices lie
Chapter 13 Conic Sections and Systems of Nonlinear Equations
26.
22
31
1
916
xy
The center of the ellipse is
3, 1 .
The vertices lie 4
units above and below the center. The endpoints of the
minor axis lie 3 units to the right and left of the center.
Center Vertices
Endpoints of
Minor Axis
27.
2
2
21
25 36
y
x
The center of the ellipse is
0, 2 .
Because the
denominator of the
2
termy
is greater than the
Center Vertices Endpoint
Minor Axis
0, 2
0, 2 6
0, 4
05,2
5, 2
28.
22
41
425
xy
The center of the ellipse is
4,0 .
The vertices lie 5
units above and below the center. The endpoints of
the minor axis lie 2 units to the left and right of the
center.
29.
2
2
321
9
xy
denominator of the
2
term,y
the major axis is
horizontal. Since
2
9,a
3a
and the vertices lie
3 units to the left and right of the center. Since
2
Center Vertices
Endpoints of
Minor Axis
3, 2
33,2
3, 2 1
Section 13.2 The Ellipse
30.
2
2
22
231
16
23
1
16 1
xy
xy
The center of the ellipse is
2,3 .
The vertices lie
4 units to the left and right of the center. The
endpoints of the minor axis lie two units above and
below the center.
Center Vertices
Endpoints of
31.
22
22
914 336
914 3 36
xy
xy
units above and below the center. Since
2
4,b
2b
and endpoints of the minor axis lie 2 units to
the right and left of the center.
1, 0
3, 3
22
4, 3 6
4,3
41,3
3, 3
Chapter 13 Conic Sections and Systems of Nonlinear Equations
33. From the graph we see that the center of the ellipse
is
,1,1hk
. We also see that the major axis is
horizontal. The length of the major axis is 4 units,
41
34. From the graph we see that the center of the ellipse
is
,1,1hk
. We also see that the major axis
35.
22
1xy
and
22
99
xy
5
y
(0, 1)
36.
22
25xy
and
22
22
25 25
25 25
xy
xy
units (
2
25a
, so
5a
), and horizontal minor
axis of length 2 units (
2
1b
, so
1b
).
y
(0, 5)
Check each intersection point.
(0, 3)
5
y
Section 13.2 The Ellipse
38.
22
1
436
xy
and
2x
The first equation is for an ellipse centered at the
origin with vertical major axis of length 12 units
(
2
36a
, so
6a
) and horizontal minor axis of
39.
22
22
44
44
444
xy
xy
and
22
22
22
xy
yx
yx
(1, 0)
5
y
40.
22
22
22
44
44
444
1
14
xy
xy
xy
and
3
3
xy
yx
The two graphs never cross, so there are no
intersection points.
We want to graph the bottom half of an ellipse
centered at the origin with a vertical major axis of
length 8 units (
2
16a
, so
4a
) and horizontal
minor axis of length 4 units (
2
4b
, so
2b
).
Chapter 13 Conic Sections and Systems of Nonlinear Equations
42.
2
2
22
22
22
44
44
44
44
yx
yx
yx
xy
43. From the figure, we see that the major axis is
horizontal with a = 15, and b = 10.
22
22
22
1
15 10
1
225 100
xy
xy
Since the truck is 8 feet wide, we need to determine the
height of the archway at
84
2
feet from the center.
22
41
225 100
y
44. From the figure, we see that the major axis is
horizontal with a = 25, and b = 20.
22
22
22
1
25 20
xy
xy
2
2
64 400
336
336 18.3
y
y
y
The height of the archway 10 feet from the center is
approximately 18 feet. Since the truck is 14 feet
high, the truck will clear the archway.
45. a.
22
22
22
1
48 23
1
2304 529
xy
xy