Chapter 12
CONIC SECTIONS
12.1 Introduction to Conics;
the Parabola
Exercises
2. When the equation of a parabola is given in
6.
(
)
()
()
()
2
2
11
11
1, 1
yx
yx
hk
=− +
=− +
==
Vertex:
(
)
1,1
Axis of symmetry: 1x=
() ()
() ()
2
2
11 ,
2 2 1 1 10 2,10
xyx xy=− +
−−−+= −
8.
()
()
()
()
()
2
2
131
3
131
3
3, 1
Vertex: 3, 1
Axis of symmetry: 3
yx
yx
hk
x
=− + −
=− − − + −
=− =−
−−
=−
() ()
2
333
1
333113,1
3
−−−+−=− −−
()
()
() ( )
2
2
2
144
2231 2,
333
177
1131 1,
333
1
003140,4
3
⎛⎞
⎟
⎜
−−−+−=− −−
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
−−−+−=− −−
⎟
⎜⎟
⎜
⎝⎠
−+−=− −
Chapter 12 Conic Sections
242
10. (continued)
()
() ( )
2
2
28 ,
3 2 3 8 10 10,3
yxy xy=−
−=
12.
2
41
xyy
=− − +
()
() () ( )
2
2
2
41 ,
5545144,5
4444111,4
yxyy xy=− − +
−−−−−+=− −−
−−−−−+= −
12. (continued)
()
2
22
2
329
22
32 93
22
yxx
yxx
=−−
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=−+ −−
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠
⎜⎟
⎜
⎝⎠
(
)
()
2
3112
1, 12
Vertex: 1, 12
Axis of symmetry: 1
yx
hk
x
=−−
==−
−
=
()
2
369 ,
xyxx xy=−−
Section 12.1 Introduction to Conics; The Parabola
243
16.
()
2
2
481
421
xyy
xyy
=− + −
=− − −
()
() () ( )
() () ( )
2
2
2
481 ,
242 821 33 33,2
1 4 1 8 1 1 13 13, 1
yxyy xy=− + −
−−−+−−=− −−
−−−+−−=− −−
18. 2
22
2
35
33
35
22
yx x
yx x
=+−
⎛⎞ ⎛⎞
⎟⎟
⎜⎜
=++ −−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
20.
()
2
2
456207
414207
yx x
yx x
=++
=++
22.
()
2
2
2
2
336136
3 12 136
12
xy y
xyy
=− + −
=− − −
⎛⎞
⎛⎞
−⎟
⎜⎟⎟
24.
(
)
2
2
665
65
xy y
xyy
=−+
=−+
22
2
11
656
xyy
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=−+ +−
71 71
, Vertex: ,
22 22
hk ⎛⎞
⎟
⎜
== ⎟
⎜⎟
⎜
⎝⎠
Chapter 12 Conic Sections
244
28.
(
)
2
5, 0
05
hk
xay
=− =
=−−
Passes through
(
)
1, 2−−
30.
()
()
2
2
22
2
42440
4640
66
46 404
22
xy y
xyy
xyy
=− + −
=− − −
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− − + − − −
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠
⎜⎟
⎜
⎝⎠
32.
()
()
()
()
2
2
132
2
132
2
3, 2
Vertex: 3, 2
Axis of symmetry: 3
yx
yx
hk
x
=++
=−−+
=− =
−
=−
() ()
2
132 ,
2
xy x xy=++
32. (continued)
34.
()
()
2
2
22
2
246
226
22
22 62
22
Axis of symmetry: 1
yxx
yxx
yxx
x
=− + −
=− − −
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− − + − − −
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠
⎜⎟
⎜
⎝⎠
=
()
() () ( )
() () ( )
() () ( )
() () ( )
() () ( )
2
2
2
2
2
2
246 ,
222 426 22 2,22
121416 12 1,12
02040660,6
12141641,4
22242662,6
xyxx xy=− + −
−−−+−−=− −−
−−−+−−=− −−
−+−=− −
−+−=− −
−+−=− −
Section 12.2 The Circle
245
38. a.
(
)
()
()
()()
(
)
2
2
2
2
10 (8 2 )
80 12 2
2680
2698029
2398
Axx
Axx
Axx
Axx
Ax
=− +
=+ −
=− − +
=− − + + − −
=− − +
40.
(
)
()
The vertex is at 0,4 and the parabola
1120
passes through ,116 560,116 .
2
⎛⎞
⎟
⎜=
⎟
⎜⎟
⎜
⎝⎠
Mindstretchers
1. Answers may vary.
2. Point: a horizontal plane that intersects the
cone through the vertex.
Line: a slanted plane that intersects the cone
12.2 The Circle
Exercises
2. The midpoint of a line segment with
endpoints
(
)
11
,xy and
()
22
,xy is
8.
()
()
() ( )
2
2
22
72 48
512
25 144
d=−+−−
=+−
=+
12.
()
()
()
()()
22
22
97 810
22
44
8
d=−−− +−
=− +−
=+
=
Chapter 12 Conic Sections
246
16.
22
51 13
66 44
d⎛⎞⎛⎞
⎟⎟
⎜⎜
=−+−
⎟⎟
⎜⎜
⎟
⎟⎜
⎜⎝⎠
⎝⎠
18.
()
()
(
)
()
()
()
2
2
22
25 5 36 6
546
5166
596
101 10.0 units
d=−+−−
=+
=+
=+
=≈
)
24.
(
)
12 5 910 171
,,
22 22
⎛⎞
⎛⎞−+− −+ ⎟
⎜⎟
⎜
⎟=−
⎜⎟
⎜
⎟⎟
⎜⎜
⎟
⎜⎝⎠
⎝⎠
⎣
30.
(
)
(
)
92 32 55 55
,
22
⎛⎞
−+− +− ⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎟
⎜
34.
(
)
(
)
() ()
22
22
2
149
143
xy
xx
+++ =
⎡⎤⎡⎤
−− + −− =
⎣
⎦⎣ ⎦
Section 12.2 The Circle
247
38.
22
22
412
412
xy x
xxy
+−=
−+=
40.
22
22
4630
xy xy
+++−=
⎣⎦⎣⎦
(
)
Center: 2, 3 ; radius: 4−−
42.
22
22
4210
421
xy xy
xxyy
++++=
+++ =−
48. The radius is the distance from the center to
the point through which the circle passes.
()
()
()
22
32 58
r=−−− +−
Chapter 12 Conic Sections
248
50. The center is the midpoint of the diameter.
(
)
()
75
73 102
Center = , , 5,1
22 22
⎛⎞
⎛⎞+−
+⎟
⎜⎟
⎜
⎟==
⎜⎟
⎜
⎟⎟
⎜⎜
⎟
⎜⎝⎠
⎝⎠
The radius is half the length of the diameter.
52. 22
840xy xy+−−+=
22
84
xxyy
−+−=−
54.
22
22
14 2 0
14 2 0
xy xy
xxyy
++ −=
++−=
56.
()
22
22
22
2 2 20 10 0
21010
10 5
xy y
xy y
xy y
+++=
++ =−
++ =−
() ( )
222
124
xy
⎡⎤
−− + − =
⎣
⎦
(
)
Center: 1, 2 ; radius: 4−
22
22
36
36
22
xx y y
⎛⎞ ⎛⎞
−⎟⎟
⎜⎜
−+ + + +
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
Section 12.2 The Circle
249
62.
()
()
()
22
6.8 3.8 2.9 3.1
d=− −− +− −
64.
(
)
(
)
4.8 9.4 0.7 5.3 14.2 6
,,
2222
7.1, 3
⎛⎞
⎛⎞−+− +−
⎟
⎜⎟
⎜
⎟=
⎜⎟
⎜
⎟⎟
⎜⎜
⎟
⎜⎝⎠
⎝⎠
=−
66. The sales representative drove a total of
20 25 45+= mi east and 17 7 24+=
)
)
)
)
68. a. The midpoint between
(
)
(
)
3,5 and 9,7 is
()
3957 1212
,,6,6.
22 22
⎛⎞⎛⎞
++
⎟⎟
⎜⎜
==
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠⎝⎠
The slope of the line segment is
)
)
b. Find the x-intercept by substituting
0.y=
)
0324
x
=− +
68. c.
(
)
(
)
Find the distance from 8,0 to 3,5 .
52
=
(
)
(
)
Find the distance from 8,0 to 9,7 .
()()
() ( )
22
22
98 70
17
149
d=−+−
=+
=+
70. Diameter = 42 ft; radius = 21 ft
22 2
22
21
441
xy
xy
+=
+=
72. a.
(
)
Center: 2, 6 ; radius: 5−
Chapter 12 Conic Sections
250
Mindstretchers
1. Let P represent
(
)
11
,xy, let Q represent
(
)
22
,xy , and let M represent the midpoint
1212
,
22
xxyy
⎛⎞
++
⎟
⎜⎟
⎜⎟
⎜
⎝⎠
.
()( )
()( )
22
21 21
22
21 21
1
4
1
2
xx yy
xx yy
⎡⎤
=−+−
⎢⎥
⎣⎦
=−+−
)
)
)
)
22
12 12
22
22
xx yy
MQ x y
⎛⎞⎛ ⎞
++
⎟⎟
⎜⎜
=− +−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠⎝ ⎠
.PM MQ=
2. The center of the middle circle is
(
)
,0r
and its radius is 2 .r The center of the
largest circle is 2r units to the right of
)
)
)
)
)
3. a. No, y is not a function of .x The
equation is the equation of a circle,
and since the graph of a circle does
not pass the vertical-line test, it is not
a function.
b.
(
)
(
)
22
2
24 5
yx
−=−−
12.3 The Ellipse and
the Hyperbola
Exercises
2. If an ellipse has center
(
)
0,0 and passes
through
(
)
(
)
(
)
(
)
,0 , ,0 , 0, , and 0, ,aa b b−−
y-intercepts:
(
)
(
)
0, 3 and 0,3−
Section 12.3 The Ellipse and the Hyperbola
251
8.
22
)
)
)
)
1
16 49
xy
+=
10. 22
22
)
)
)
)
)
)
)
)
436 36
436 36
36 36 36
xy
xy
+=
+=
12. 22
22
2 8 128
28 128
xy
xy
+=
x-intercepts:
(
)
(
)
8, 0 and 8, 0−
y-intercepts:
(
)
(
)
0, 4 and 0, 4−
Chapter 12 Conic Sections
252
16.
22
22
22
1
25 4
1
52
xy
xy
−=
−=
18.
22
22
22
1
49 9
1
73
yx
yx
−=
−=
3, 7ab==
20. 22
22
22
464
464
64 64 64
xy
xy
−=
−=
22. 22
22
28 200
yx
−=
Section 12.3 The Ellipse and the Hyperbola
253
24. 22
66 18
xy
+=
26.
(
)
2
2
2
927
927
9 0 +27 Parabola
yx
xy
xy
+=
=− +
=− −
30. 6, 13ab==
22
22
22
1
613
1
36 169
xy
xy
+=
+=
34.
9
99 7
7
b
ba
aa
=
==⇒=
36. (continued)
()
2
2
2
32 9
23
3
203 Parabola
3
yx
yx
yx
=−
=−
=−−
42. 22
22
22
12 3 108
12 3 108
108 108 108
xy
xy
+=
+=
Chapter 12 Conic Sections
254
42. (continued)
44. a. 4225 65 2 2 65 130
2500 50 2 2 50 100
aa
bb
==⇒=⋅=
==⇒=⋅=
The length is 130 yd and the width is
100 yd.
)
)
46. a. 22
22
4 6400
4 6400
6400 6400 6400
yx
yx
−=
−=
b. The graph is increasing, with y-intercept
(
)
0,80 .
48. a. 70 150
35, 75
ab== = =
48. b.
Mindstretchers
1. As the tacks are moved closer together,
the shape of the ellipse becomes more
nearly circular. As the tacks are moved
2.
2
bb b b
maa a a
−− −
===
−− −
The equation of the line is:
b
()
()
a
b
yb xa
a
b
yxab
a
b
−=− +
=− + +