Chapter 12
the Parabola
Exercises
2. When the equation of a parabola is given in
the form 2,yax bxc=++ it can be written
4. A parabola that opens to the left or to the
()
1, 1
hk
==
Vertex:
(
)
1, 1
Axis of symmetry: 1x=
() ()
2
11 ,
xyx xy=− +
()
()
3
3, 1
Vertex: 3, 1
Axis of symmetry: 3
hk
x
=− =−
−−
=−
() ()
2
131 ,
xy x xy=− + −
()
()
2
2
2
144
2231 2,
333
177
1131 1,
333
1
⎛⎞
⎟
⎜
−−−+ −=− −−
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
−−−+ −=− −−
⎟
⎜⎟
⎜
⎝⎠
10.
28
xy
=−
242
10. (continued)
()
() ( )
() ( )
2
2
2
28 ,
3 2 3 8 10 10,3
22280 0,2
yxy xy=−
−=
−=
12.
()
()
22
2
2
44
41
22
25
5, 2
xyy
xy
hk
⎛⎞
⎛⎞ ⎛⎞
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− + + + +
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠
⎜⎟
⎜
⎝⎠
=− + +
==−
()
22
2
22
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
() () ( )
() () ( )
() () ( )
() () ( )
2
2
2
2
131 6190 1,0
0306099 0,9
1 3 1 6 1 9 12 1, 12
2326299 2,9
−−−−−=−
−−=− −
−−=− −
−−=− −
243
16.
()
22
481
xyy
=− + −
()
() () ( )
() () ( )
2
2
481 ,
1 4 1 8 1 1 13 13, 1
0408011 1,0
yxyy xy=− + −
−−−+−−=− −−
−+−=− −
18. 2
2
2
35
9209
3444
329
24
yx x
yx x
yx
=+−
=++−−
⎛⎞
⎟
⎜
=+ −
⎟
⎜⎟
⎜
⎝⎠
()
()
22
22.
()
2
2
336136
12
312 2
xy y
xyy
=− + −
⎛⎞
⎛⎞
−⎟
⎜⎟⎟
⎜
⎜
=− − + ⎟⎟
⎜
⎜⎟⎟
⎜
⎝⎠
⎜⎟
⎜
⎝⎠
24.
(
)
2
665
65
xy y
xyy
=−+
=−+
22
11
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
71 71
, Vertex: ,
hk ⎛⎞
⎟
⎜
== ⎟
26. 1, 8hk==
(
)
(
)
2
18
Passes through 1,6
yax=−+
−
()
2
6118
a
=−− +
244
28.
(
)
2
5, 0
05
hk
xay
=− =
=−−
Passes through
(
)
1, 2−−
()
2
1205
a
−= −− −
()
22
⎛⎞
⎛⎞ ⎛⎞
32.
()
()
()
2
3, 2
Vertex: 3,2
Axis of symmetry: 3
hk
x
=− =
−
=−
() ()
() ()
2
2
132 ,
2
1
553245,4
xy x xy=++
−−++=−
32. (continued)
34.
()
22
246
226
yxx
=− − −
⎛⎞
() () ( )
() () ( )
() () ( )
() () ( )
() () ( )
() () ( )
2
2
2
2
2
2
222 426 22 2,22
121416 12 1,12
02040660,6
12141641,4
22242662,6
323436123,12
−−−+−−=− −−
−−−+−−=− −−
−+−=− −
−+−=− −
−+−=− −
−+−=− −
36.
(
)
1,16 ; it shows that the object will reach
245
(
)
()
()
()()
2
2
2698029
Axx
=− − + + − −
714 98
×=
40.
(
)
The vertex is at 0,4 and the parabola
()
112 313,600
a
=
vertex.
3. The x-coordinate of the vertex is 3.
2
2. The midpoint of a line segment with
endpoints
(
)
11
,xy and
()
22
,xy is
6. The equation 22
() ( )
22
512
=+−
10.
()
()
43 512
5 2 7.1 units
d=−+−
=≈
()
()
() ( )
22
112
145 12.0 units
=+−
=≈
Chapter 12 Conic Sections
246
66 44
⎜⎜
⎟
⎟⎜
⎜⎝⎠
⎝⎠
64
32
41
⎜
⎝⎠ ⎝⎠
()
(
)
()
()
()
22
546
5166
101 10.0 units
=+
=+
(
)
4.5, 3
=−
28.
,,
2222
=⎜
⎟⎟
⎜⎟⎟
⎜
⎝⎠⎝⎠
30.
()
,
22
22
⎜⎟
⎜⎟
⎜⎟
⎟
⎜
⎝⎠
⎟
⎜⎟
⎟
⎜
⎝⎠
32.
()()
22
2
004
xy
−+−=
() ()
143
xx
⎡⎤⎡⎤
−− + −− =
⎣
⎦⎣ ⎦
() ()
22
332
xy
⎡⎤
−+−− =
⎣
⎦
247
38.
22
22
22
412
412
44
xy x
xxy
+−=
−+=
⎛⎞ ⎛⎞
−−
40.
22
22
46
(
)
Center: 2, 3 ; radius: 4−−
22
22
22
4210
421
42
xxyy
+++ =−
⎛⎞ ⎛⎞
44.
() ()
6108
xy
−+−− =
46.
() ( )
(
)
102
xy
⎣⎦
48. The radius is the distance from the center to
the point through which the circle passes.
()
()
()
2
2
13
19
10
=− +−
=+
=
248
22 22
⎜
⎟⎟
⎜⎜
⎟
⎜⎝⎠
⎝⎠
The radius is half the length of the diameter.
22
2
410
2
=
=
52. 22
22
22
22
22
84
81
822
81
xxyy
xx yy
−+−=−
⎛⎞ ⎛⎞
⎟⎟
⎜⎜
−+− +−+−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
⎛⎞⎛⎞
⎜⎟
⎜
54.
22
22
22
14 2 0
14 2
14 2
22
xxyy
xx yy
++−=
⎛⎞ ⎛ ⎞
−
⎟⎟
⎜⎜
++ +−+
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝ ⎠
56.
()
22
22
2 2 20 10 0
10 5
xy y
xy y
+++=
++ =−
⎛⎞ ⎛⎞
58.
22
22
22
2411
24
24
22
xxyy
xx yy
++−=
⎛⎞ ⎛ ⎞
−
⎟⎟
⎜⎜
++ +−+
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝ ⎠
(
)
Center: 1, 2 ; radius: 4−
60. 22
22
22
36
36
22
36
922
xx y y
−⎟⎟
⎜⎜
−+ + + +
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
⎛⎞⎛⎞
−⎟⎟
⎜⎜
=− + +
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠⎝⎠
249
()
(
)
7.1, 3
66. The sales representative drove a total of
20 25 45
+= mi east and 17 7 24+=
mi south to the state college.
68. a. The midpoint between
(
)
(
)
3,5 and 9, 7 is
(
)
(
)
636
yx
−=− −
0.y=
0324
x
=− +
149
=+
22
72. a.
(
)
Center: 2, 6 ; radius: 5−
65 8.1
=≈
250
1. Let P represent
(
)
11
(
)
22
,xy , and let M represent the midpoint
121 2
,
22
xxyy
⎛⎞
++
⎟
⎜⎟
⎜⎟
⎜
⎝⎠
.
22
12 12
11
21 21
22
4
xx yy
PM x y
⎛⎞⎛ ⎞
++
⎟⎟
⎜⎜
=−+−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠⎝ ⎠
⎢⎥
⎣⎦
()( )
22
22
21 21
44
4
xx yy
=−+−
⎢⎥
⎣⎦
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
(
)
,0 ,r at
(
)
3,0,r and it has a radius of 4 .r
(
)
(
)
(
)
222
304
xr y r
−+−=
not pass the vertical-line test, it is not
a function.
b.
(
)
(
)
()
()
22
2
2
24 5
24 5
24 5
yx
yx
yx
−=−−
−=± − −
=± − −
the Hyperbola
2. If an ellipse has center
(
)
4. If a hyperbola has center
(
)
a and
b is 1.
6.
1
25 9
+=
(
)
(
)
251
8.
22
22
1
16 49
1
xy
+=
+=
10. 22
22
436 36
1
91
1
xy
xy
+=
+=
22
28 128
128 128 128
xy
+=
14. 22
22
18 2 162
162 162 162
xy
+=
252
16.
22
22
1
25 4
1
52
xy
−=
18.
1
49 9
3, 7ab==
(
)
(
)
-intercepts: 0, 7 and 0, 7
77
y
−
22
22
464
64 64 64
xy
xy
−=
22. 22
22
22
28 200
28 200
200 200 200
yx
yx
−=
−=
253
24. 22
22
66 18
66 18
xy
xy
26.
927
yx
+=
28. 22
22
10 12 120
yx
30. 6, 13ab==
22
xy
32. 43, 5ab==
()()
34.
9
99 7
7
b
ba
aa
=
==⇒=
36. 22
22
28 72
28 72
xy
xy
−=
6, 3ab==
2
32 9
23
yx
=−
b
=
1
yx
−=
22
12 3 108
108 108 108
xy
+=
3, 6ab==
x-intercepts:
(
)
(
)
3, 0 an d 3, 0−
254
2500 50 2 2 50 100
bb
==⇒=⋅=
46. a. 22
22
4 6400
6400 6400 6400
yx
−=
b. The graph is increasing, with y-intercept
48. a. 70 150
35, 75
22
ab== = =
1. As the tacks are moved closer together,
2
−− −
The equation of the line is:
()
b
yb xa
−= −
(
)
(
)
2.
bb b b
−− −
()
a
b
a
b
yxbb
=− − +
3. Answers may vary.
255
Systems of Equations
Exercises
(2)
35
yx
=+
Use substitution.
2
2
73 5
320
xx
xx
+= +
−+=
(2)
4
=−
Use substitution.
(
)
2
22
42 2
42
416
816 16
70
yy
yy y
yy
−+=
−++=
−=
2
212
6
6
y
y
y
=
=
=±
Solve equation (2) for x, then substitute the
expression for x in equation (1) and solve for y.
55xy x y+=⇒=−
(
)
(
)
22
2
525
2100
250
yy
yy
yy
−+=
−=
−=
20 50
yy
=−=
expression for x in equation (1) and solve for x.
12
2
xyyx=⇒=
2
2
23 24
3440
xx x
xx
=−−
−−=
(2)
6
=+
Use substitution.
(
)
2
22
684
yy
++=
256
22
22
30 160
316
316
yy
yy
yy
−= + =
==−
=± =± −
2
(2)
515
9 45
x
−=
=
2
2
2
45 30
20 30
10
y
y
y
⋅+ =
+=
=
2
(2)
432 432
3 12
y
+=⎯⎯→+=
=
2
420
16
x
x
+=
=
18.
()
1
22 22
2
28 40 28 40
5 8 16 5 8 16
3 24
xy xy
xy xy
x
×−
+=⎯⎯→+=
+=⎯⎯⎯→− − =−
−=
7
y
=±
Solutions:
22, 7, 22,7,
ii
−− −
20.
22 22
2
2
42 32 42 32
4 16
xy xy
y
−=⎯⎯→−=
=
10
10
x
x
=
=±
22. 22 22
3
22 22
3 3 108 3 3 108
272 63216
xy xy
xy xy
×
−=⎯⎯→−=
+=⎯⎯→+=
257
2
236 72
0
y
y
⋅+ =
=
24.
22
24 22
4
22 2 2
39
16424
46
9 3 27 36 12 108
xy xy
xy x y
×
×
+=⎯⎯→ + =
+=⎯⎯→+=
2
12 4 24
412
y
y
+=
=
(
)
(
)
(
)
(
)
26.
9
5
y
=±
22
2
416
45 16
xy
x
−=
−⋅=
(2)
Use substitution.
()
(
)
70
xx
+=
22
2
2
070
07
02 7
xx
xx
yx
=+=
==−
=+ =± −
30.
22 22
321 321
xy xy
+=⎯⎯→+=
(
)
(
)
22
213213
33
yy
yy
+= −+=
=± =±
(2)
3
yx
=− +
22
2
2
73
210
xx
x
−=− +
=
258
34.
22
18 22
2
2
1618
18 3
3 18
6
6
xy xy
x
x
x
×
×
+=⎯⎯→ + =
−=−
=
=±
22
2
618
66 18
xy
y
+=
+=
2
0.5 2.5 2 0
540
tt
tt
−+=
−+=
and 2004.
38.
22
4 10, 000
2000
xy
x
+=
=
The pathways intersect at the points
(
)
(
)
20 5, 20 5 , 20 5, 20 5 ,
−− −
40. Let
()
,hk be the coordinates of point C.
The distance from C to
(
)
0, 0 is 41.
()()
(
)
2
22
22
0041
41
hk
hk
−+− =
+=
The distance from C to
()
3,1 i s 13.
(
)
2
2
19 3 41
hh
+− =
5, 57, 160ab c==− =
()
25
5, 6.4
h
h
=
=
19 3 19 3 6.4 19 19.2 0.2
kh
=−=−⋅ =− =−
259
2. 22
416
xy
−=
2
Solving the system gives us
22
416
2
xy
yx
−=
=
(3)
22
22
2
2
222
313
5 35
7
xy
xy
x
x
+=
−=
=
=
Inequalities and Nonlinear
Systems of Inequalities
Exercises
(
)
Boundary is solid. Test point is 0, 0 .
?
22
?
004
04 False
+≥
≥
(
)
Shade the region not containing 0,0 ,
260
8.
(
)
(
)
()()
(
)
22
321
The corresponding equation is:
321
Circle with radius = 1, center: 3,2
xy
xy
++−<
++− =
−
()
22
2
28
44
yx x
=−
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
The corresponding equation is:
22
22
16 64
1
64 4
xy
xy
+=
+=
14. 22
25 9 225xy+≤
22
22
25 9 225
xy
+=
261
16. 22
2832yx−<
The corresponding equation is:
22
22
28 32
1
16 4
yx
yx
−=
−=
()
()
2, 4
A hyperbola passing through 0, 4 and
4
0, 4 with asymptotes or
2
2 and 2
ab
yx
yx y x
==
−
−=±
==−
Boundary is dashed.
There are three regions.
(
)
Region I, the uppermost region.
Test point is 0,10 .
The corresponding equation is:
yx≤− +
(
)
2
2
5
5
yx
yx
=− +
=− +
(
)
()
() ( )
() ( )
() ( )
() ( )
() ( )
() ( )
2
2
2
2
2
2
2
Parabola opening down, vertex at 0,5
5,
33543,4
22512,1
11541,4
00550,5
11541,4
22512,1
xyx xy=− +
−−−+=− −−
−−−+= −
−−−+= −
−+=
−+=
−+=
262
(
)
(
)
Circle, center 0,0 , radius 4.
Boundary is dashed. Test point is 0,0 .
?
22
?
0016
016 False
+>
>
Shade the area not containing (0,0),
the area outside the circle.
22
4
xy
+<
22. 221yx x<++
The corresponding equation is:
(
)
2
221 1 0yx x y x=++⇒=++
(
)
Parabola opening up with vertex at 1,0−
()
2
21 ,
xyxx xy=++
Boundary is dashed. Test point is
(
)
0, 0 .
(
)
??
2
00 20 1 01 True<+ +⇒<
(
)
Shade the area containing 0,0 .
The corresponding equation is:
(
)
22
22
2
24 2 4
22
24
22
yx x y x x
yxx
=− − − ⇒ =− + −
⎛⎞
⎛⎞ ⎛⎞
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− + + − +
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠⎜⎟
⎜
⎝⎠
Boundary is dashed. Test point is
(
)
0, 0 .
263
The corresponding equation is:
22
12 27 108
xy
+=
(
)
() ()
?
22
?
12 0 27 0 108
0108 False
+>
>
(
)
Shade the area not containing 0,0 ,
the outside of the ellipse.
22
25 25xy+<
The corresponding equation is:
22
22
25 25
xy
xy
+=
26. 22
The corresponding equation is:
22
16 4 64
xy
+=
Boundary is dashed. Test point is
(
)
0, 0 .
() ()
?
22
?
16 0 4 0 64
064 False
+>
>
(
)
Shade the region not containing 0,0 ,
the region outside the ellipse.
22
936yx−≤
The corresponding equation is:
22
22
936
yx
yx
−=
264
28. 223xy y>−+
The corresponding equation is:
()
(
)
2
12
xy
=− +
(
)
Parabola opening right, vertex at 2,1
()
2
23 ,
yxyy xy=−+
Boundary is dashed. Test point is
(
)
0, 0 .
()
?2
?
00 20 3
03 False
>− +
>
(
)
Shade the area not containing 0,0 .
223xy y≥− + −
The corresponding equation is:
2
23
xy y
=− + −
() () ( )
() () ( )
2
1121366,1
0020333,0
−−−+−−=− −−
−+ −=− −
Boundary is solid. Test point is
(
)
0, 0 .
The combined areas, not containing
right opening parabola.
The corresponding equation is:
()
2
22
2
2
23
22
23
22
14
yx x
yxx
yx
=− + +
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− − + + +
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠⎜⎟
⎜
⎝⎠
=− − +
(
)
Parabola opening down, vertex at 1,4 .
()
2
23 ,
xyxx xy=− + +
Boundary is solid. Test point is
(
)
0, 0 .
() ()
?2
00203
≤− + +
265
The corresponding equation is:
22
22
936
1
436
xy
xy
−=
−=
2, 6
Hyperbola passing through
ab
==
(
)
34. 22
The corresponding equation is:
()
(
)
22
4
Circle with radius 2, center 0,0 .
Boundary is solid. Test point is 0,0 .
xy+=
?
22
004
+≤
0.3 9Cx≥+
Boundary is solid. Test point is
(
)
0, 0 .
(
)
0, 0 .
266
()
()
12 0.2
0.2 30 180
Rx
=− − +
() () ( )
60 12 60 0.2 60 0 60 , 0
−=
Boundary is solid. Test point is
(
)
10,0 .
()
?
?
0 120 0.2 100
0120 20
0 100 True
≤−
≤
The corresponding equation is:
(
)
20, 15
ab==
()()( )()
Ellipse passing through
25,0 , 25,0 , 0, 20 , and 0,20
−−
?
1
625 400
01 True
+≤
≤
(
)
Shade the region containing 0, 0 ,
267
1. Yes, the system
2
2
(
)
0, 0 .
Note that
(
)
2
239xy++ = and
2. 22
936xy+≥
The corresponding equation is:
22
22
936
xy
xy
+=
The corresponding equation is:
1
16 4
−=
21 1
,or and
42 2
yxyxyx
=± = =−
24
2
xy
−=
()
12,
xy x xy=−
?
04 False
>
(
)
Shade the region not containing 0, 0 .
3. The region is inside the ellipse whose
so 3, 5.ab==
R
) must hold for
(
)
0, 1 .−
22
?
268
3. (continued)
The region is below the parabola whose
vertex is
(
)
0,4 and passes through (2,0).
The boundary is solid.
(
)
2
04
yax
=−+
R
) must hold for
(
)
0, 1 .−
24.xy+≤
The region is below the dashed line. The line
passes through the points
(
)
(
)
0,0 and 1, 1 .−
(
)
the relation (
?
R
) must hold for
(
)
0, 1 .−
The three inequalities are:
22
2
1
925
4
xy
xy
+≤
+≤