ht
1O-2- Gircles Extra Practice
49-58, write the standard form of the equation and the general form of the equation of each circle
(h, k). By hand, graph each circlecenter
radius r and
50. r : 2; (h, k) : (-2,1')
49. r : 1,; (h, k) = (1, -1)
In Problems
51. r = 2; (h, k) : (0,2)
53. r : 5; (h, k) = (4, -3)
55. r = 2;(h, k): (0,0)
57. r:l;(h, k) =
52. r = 3; (h, k) = (1' 0)
54. r : 4; (h, k) : (2, -3)
56. r = 3; (ft, /c) = (0,0)
(1,01
58. r = j;(n,
D: (o'-)
In Problems 5948, find rhe center (h, k) and radius r of each circle. By hand, graph each circle.
(y-1)2=r
62. (x + L)2 + 0/ - t)2 =2
64. x2+y2-6x+2y+9:0
66. f +y2+x+v-t:0
68. 2f + 2y2 + 8x + 7 = 0
x2+y2:4
6t. (x -3)2 + y2:4
63. x2+yz+4x-4y-l=0
6s. f +y2-x*2y+1:0
67. 2* + 2y' - 12x + 8y - 24:0
60. x2+
59.
In Problems 69-74, find the general form of the equation of each circle.
69. Center at the origin and containing the point (-2,3)
70. Center (-3,1) and tangent to the y-axis
7l-. Center (1,0) and containing the point (-3,2)
72. Wirh endpoints of a diameter at (1,4) and (-3, 2)
73. Center (2,3) and tangent to the.r-axis
74. With endpoints of a diameter at @,3) and (0, 1)
1O-3
-
ElliPses Extra Practice
In Problems 9-18, find the uertices and foci of each ellipse. Graph each equation
s.*.I:'
10.T.i:'
11.t.*=' D.i.*='
+f :16
t7. *+f :rc
f +vf :n
18. f +f :+
ffi. 4f +f:8
13. 4f
14.
16. $P+9f
=t\
In Problems 19-28, find an equation for each ellipse.
D. Center at (0, 0); focus at (3, 0); vertex at (5, 0)
2L. Center at (0,0); focus at (0, -4); vertex at (0,5)
23.: Foci at ('r2,0); tength of the major axis is 6
25. Foci at (0, *3); x-intercepts arc !2
n. Center at (0, 0); vertex at (0, a); b : 1
20. Center at (0,0); focus at (-1,0); vertex at (3,0)
?2. Center at (0,0); focus at (0, L); vertex at (0, -2)
24. Focus at (0, -4); vertices at (0, +8)
26. Foci at (0, t2); length of the major axis is 8
28. Vertices at (+5, 0): c = 2
of
Pg.A
In Problems 3344, find the centei, foci, and
ii
of each ellipse Graph each equation
(x + 4)2 (v + 2\2
uertices
(;--3\2_Lv+l)2_,
_I
49
34.
3s.
(r + 5f + 4A.- 4)2 = rc
37, f +4x+4y2-8y*4=0
39. zi+zf-ar+6y+5=0
4L Sf++f-18r+16y-11:0
43. 4*+f+4y:s
g
L
"
4
'
-1
-L
36.
9(x-3)2+(.y+2)2:ts
38.
*+zf-r2y+9=o
40.
4x2+3y2+8r-6y=S
a. *+ef+6x-18y+9:o
4. ef+f-1&:o
In Problems 45-54, find an equation for each ellipse
45.
Center at (2, -2); vertex at (7, -Z);focus at
(4,
Vertices at (4,3) and (4,9); focus at (4, g)
49.
Foci at (5, 1) and
axis is 8
(-1,
1); length of the major
Center at (1, 2); focus at (4, 2); contains the point
Center at
support a bridge that is to span a river 20 meters
wide. The. center of the arch is 6 meters above
the center of the river (see the figure). Write an
equation for the ellipse in which the x-axis
coincides with tbe water level and the y-axis
passes through the center of the arch.
Vertices at (2,5) and (2,
<.,
Center at (1,2); focus at (1, 4); contains tne poinii
62.
-1);c = Z
,,
..=
(2.2)
i
63.
a
bridge is a semiellipse with a horizontal major
axis.The span is 30 feet, and the top of the arch
is L0 feet above the major a-ris. The roadway is
horizontal and is 2 feet above the top of the
arch. Find the vertical distance from the roadway to the arch at 5 foot intervals along the
roadway.
Whispering Galleries A hall 100 feet in length
is to be designed as a whispering gallery. If the
foci are located 25 feet from the center, how
high will the ceiling be at the center?
Center at (1,2); vertex at (1. 4); contains rhe:
point (2,2)
Whispering Galleries Jim, standing
at
one
focus of a whispering gailery, is 6 feet from the
nearest wall. His friend is standing at the other
focus, 100 feet away. What is the length of this
65.
61-.
(-3,2); vertex at (-4,2)
50.
64.
60.
(-3,3); focus at
Foci at (1,2) and
54.
Semielliptical Arch Bridge An arch in the
shape of the upper haif of an ellipse is used to
$sndsiliptical Arch Bridge The arch of
L); vertex at
g
(I,2); vertex at (4,2); contains the
point (1, 3)
59.
(-3,
,18.
(1, 3)
53.
Center at
(-3,0)
47.
51.
6.
-2)
66.
whispering gallery? How high is its elliptical
ceiling at the center?
Semielliptical Arch Bridge A bridge is built in
the shape of a semielliptical arch. The bridge
has a span of 120 feet and a maximum height
of 25 feet. Choose a suitable rectangular coordinate system and find the height of the arch
at distances of 10,30, and 50 feet from the center.
Semielliptical Arch Bridge A bridge is built in
the shape of a semiellipticai arch and is to have
a span of 100 feet. The height of the arch, at a
distance of 40 feet from the center, is to be 10
feet. Find the height of the arch at its center.
Semielliptical Arch An arct) in the form of
half..an ellipse is 40 feet wide and 15 feet high
at the center. Find the height of the arch at intervals of 10 feet along its width.
Semielliptical Arch Bridge
An arch for
a
bridge over a highway is in the form of haif an
ellipse. The top of the arch is 20 feet above the
ground level (the major axis). The highway has
four lanes, each 12 feet wide; a center safety
strip 8 feet wide; and two side strips, each 4 feet
wide.What should the span of the bridge be (the
length of its major axis) if the height 28 feet
from the center is to be 13 feet?
Ps3
In Problems
9-18, fincl an equation for the hyperbola d.escribed.
Center at (0,0); focus at (3,0); vertex at (1,0)
9.
L1.
(0,4)
13. Foci at (-5,0) and (5,0); vertex at (3,0)
Center at (0,0); focus at (0,
-6); vertex at
1,5.
Vertices at (0,
Y
17.
:2x
-6)
and (0,6); asymptote the
Center at (0,0); focus at (0,5); vertex at (0,3)
12.
Center at (0,0); focus at (-3,0); vertex at (2,0)
ir4.
Focus at (0,6); vertices at (0,
16.
_x
line
and (0,2)
(-4,0) and (4,0); asymptote
Vertices at
18.
Foci at (0,
l:-x
In Problems 19-26, find the center, transaerse axis,
le' x2
2s-t:r
20.
23. y2 -
24.
v2
9xt
:9
v2P
L__-1
16 4-!
x2-Y2=4
In Problems 27-30, write an equation for
uertices,
lL.
ttr
-2)
foci, and a.symptotes. Graph each equation
4x2-y2=16
t)
y2-4f=16
!
26.
2x2-y2=4
^
-LJ
l=-2x
each hyperbola.
!=2x
30.
v=-2x
'\5
t\
l\
t\
t\
Y
the line
and (0, 2); asymprote the line
27.
,o
-Z)
Y=zrc
Foci at (-4, 0) and (4, 0); asynptote the
!:
line
10.
!=2x
--7
/'l
/l
/l
/t
In Problems 31-38, fi.nd. an equation for the hyperbola described.
32.
31. Center at (4, -1); focus at (7, -l); vertex at
(6,
-1)
33. Centerat(-3,-4);focusat (-3,-8);vertexat'
(-3, -2)
35.
37.
Foci at (3,7) and (7,7); vertex at
Vertices at
line (.r
(6,7)
(-1, -1) and (3, -1); asymptote the
(y + l)13
- l)tz:
Pgt
Center at
(-3,
1); focus at
(-3,
6); vertex
at
(-3,4)
34.
Center at (I,4); focus at
36.
Focus at
38.
Vertices at (1,
(-a,
(-2, D:vertex
at (0, 4)
(-4,4)
(a,2)
0); vertices at
-3)
and
and (1, L); asymptote the line
@-r)n=0+Dt3
Graph each equation
In Problems j9-52, find the center, transaerse axis, aertices, foci, and asymptotes'
(x-2\2 (v+3)
4 -=-=t
4L (y-2)'-4(x+2)2=+
44. 0-3)'-(-r+2)2:q
4il. y2 - 4x2 - 4y -8.r- 4: 0
*r=o
s0. ,t'-f
3e.
::-]t
rv * 3)2
40' - 4 -
- 2\2 ='
g
42. (x+ 4)2 -9(Y -3)z =g
45. f -y'-2x-2y-t=0
48. zf -y'*4x-r4y-4=0
5L y2-4f -1,6x-2Y-\9-0
10.5
In Problems
17-32,
find
(x
-
(x+1)2-0l+2)':q
6. yt-f-4y+4x-1'=o
49. 4f-y'-24x-4y+16:o
52. f-3yt*8r-6y+4=o
43'
Parabolas Extra Practice
the equation of the parabola described. Find the tu)o points that define the latus rectum,
(0,0)
(0,0)
18. Focus at (0,2); vertex at (0,0)
20. Focus at (-a, 0); vertex at (0,0)
X7.
19.
Focus at (4,0); vertex at
2L
Focus at
27.
28.
29.
31.
Vertex at (0,0); axis of symmetry the y-axis; containing the point (2,3)
Vertex at (0,0); axis of symmetry the -r-axis; containing the point (2,3)
Focus at (0,
-3); vertex at
(-2,0);directrix the line.r :2
23. Directrix the liney = -l ;vertex at (0,0)
25. Vertex at (2, -3):focus at (2, -5)
(-3, a); directrix the line y
Focus at
(-3, -2);
In Problems
33-50,
find
1
:2
the line x : 1
Focus at
directrix
22. Focus at (0, -1); directrix the line y =
24. Directrix the tine * = -L;vertex at (0, 0)
26. Vertex at (4, -Z);focus ar (6, -2)
30.
32.
the uertex, focus, and directrix
33. x2:4y
36. f = -ay
39. (* - 3)'= -(y + 1)
42. (*-2)'=40-3)
f+u:4y-8
,18. xz - 4x : 2y
45.
In Problems 51-58, write an equation for
&.
Focus at ( -4,4); directrix the line y
of each parabola. Graph the equation
34. !2 :8x
37. (y - 2)' :
(-y
Focus at (2,4); directrix the line x
8(.r + 1)
+ 1)' : -a@ - 2)
43. y2-4y+4x+4:0
6. y2-2y:8.r-1
49. x2_4x=y+4
each parabola.
J5.
= -4
= -2
)
V-:-lox ".
'(x
38. + 472 : f6b) + Z)
41. (y + 3)t: 8(x - 2)
4. f +6x-4y+1=0
O. y2+2v-.r:0
\,
50. y--12y=-x-l
54.
59.
55-
Satellite
Dish A satellite
56.
dish is shaped like a
66.
Searchlights A searchlight is shaped like a paraboloid of revolution. If the light source is located2 feet from the base aiong the axis of symmetry and the depth of the searchlight is 4 feet,
what should the width of the opening be?
67.
Solar
paraboloid of revolution. The signals that emanate from a satellite strike the surface of the
dish and are reflected to a single point, where the
receiver is located. If the dish is l-0 feet across
at its opening and is 4 feet deep at is center, at
what position should the receiver be placed?
60.
Constructing a TV Dish A cable TV receiving
dish is in the shape of a parabbloid of revolution. Find the location of the receiver, which is
piaced at the focus, if the dish is 6 feet across at
its opening and 2 feet deep.
61.
Constructing a Flashlight The reflector of a
flashlight is in the shape of a paraboloid of revolution. Its diameter is 4 inches and its depth is
1 inch. How far from the vertex should the light
bulb be placed so that the rays will be reflected
parallel to the axis?
62.
Constructing
a
Headlight
A
Suspension Bridges The cables
05.
Suspension Bridges The cables of a suspension bridge are in the shape of a parabola. The
towers supporting the cabie are 400 feet apart
and 100 feet high. If the cables are at a height
of 10 feet midway between the towers, what is
the height of the cable at a point 50 feet from
the center of the bridge?
Searchlights A searchlight
trate the rays of the sun at its focus, creating a
heat source. If the mirror is 20 feet across at its
opening and is 6 feet deep, where will the heat
source be concentrated?
68.
is shaped
like
a pa-
raboloid of revolution. If the light source is located? feet from the base along the axis of symmetry and the opening is 5 feet across, how deep
should the searchlight be?
Reflecting Telescopes
A
reflecting telescope
contains a mirror shaped like a paraboloid of
revolution. If the mirror is 4 inches across at its
opening and is 3 feet deep, where will the light
collected be concentrated?
of a suspen-
sion bridge are in the shape of a parabola, as
shown in the figure. The towers supporting the
cable are 600 feet apart and 80 feet high. If the
cables touch the road surface midway between
the towers, what is the h6ight of the cable at a
point 150 feet from the center of the bridge?
64.
Heal A mirror is shaped like a paraboloid of revolution and wiil be used to concen-
sealed-beam
headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus,
is 1 inch from the vertex. If the depth is to be
2 inches, what is the diameter of the headlight
at its opening?
63.
Pg5
69.
Parabolic Arch
Bridge
.A bridge is
built in the
shape of a parabolic arch. The bridge has a span
of 120 feet and a maximum height of 25 feet.
See the illustration. Choose a suitable rectangular coordinate system and find the height of
the arch at distances of 10,30, and 50 feet from
the center.
70.
Parabolic Arch Bridge A bridge is to be built
in the shape of a parabolic arch and is to have
a span of 100 feet. The height of the arch a distance of40 feet from the center is to be L0 feet.
Find the height of the arch at its center.
71.
Show that an equation of the form
Ax2*Ey:g
A+0,E+0
of a parabola with vertex at (0,0)
and axis of spnmetry the y-axis. Find its focus
and directrix.
is the equation
Prb
10.8
- Systems of Secord Desree Equations Extra Practice
In Problems 1-20, use a graphing utility to graph
each equation of the system. Then soloe the system by
finding the
intersection Points' Express your anffiI)er correct to two
ilecimat piaces. Also solae each system algebraically.
r. {t=*'+.t
fr=*+t
s. {r=!
U=2-x
e
, Ir:x2+L
-'
ly=ax+t
6.
{r=f
"'ly:6-,
3. [t=v:r:]
J'
{.,.*:i:=l 'n {*.i:,;:l
13. [x2+y2:4
rJ'
fl''-r=a
1a
r5' lx2+y2=16
\r'-zy=6
17. {x2+t2:+-
ffi. {r=:
=2x+1.
l,:x2-,
I
|
,, [*, ,.,
'L'
l*+y:
*y=4
u.{*;y='i
2s
u.
16.
lx.+y2-8
D.
u'
l,
{*.:llo.,
(*r:"
I;r-_;
20. lx2+y2:1s
[t,=x2-+
lr=6x_r3
I
lr=v;;
ly=2*++
(v=x-1.
8l'r=*r_6x+e
,
" {.,.;;=;'| *y=+
In Problems 2l-52, solue each system. (Ise any method you wish.
+ y2 = 18
-' - ,,
zL. [z*
4.
lr:r_,
7.
" [.:rl
l*=yr-zy
23.{.r,v=zx+r
l2* +
Y'=
{n.,ijl,ll:_:
,r:,
t
1
26.{,-'-,.:;=i
zB. {rrr-3xy+6yr2x+4=0 ze.
{o*-zy+sl=t2
,o [*r-+yr+t= 0
I
2x+3y= 5 '"'
I
U-3y+4:0
L"i+y2=31.
30. {::-rri:::o
3f i-ft*-zr'+5:o
32.tt-3yz+l=o
lzf- r'+2:0
s|*syr=12
27.
l2x-7yz+5:0
{y,::;=,i
";'i;;=Z
36.
{,'*o*.,}=Z {;::fr1X
33
34.
*
{',
37.
(? 3+1=o
lZ-4+3=o qlil
3e.1."{;"_
,
| ,-
(r 1_,
43. {x2 -
li.i:-
['y - P + i:
I
y3
=26
48. {f
l.u+Y=
sl. I
2
log,y=3
llog,(ay) =
s
-+
L
"'+ xy*
+
49.
E.,
y2=
llog,(2y\ =
llog.(ay) :
3
2
=
_:
44. {*t
47.
g
t + x2 - x - 2:
1lf y+I+-=0
x-2
IL'Y
{^*:':,,,
- *, - 2yt =b
L xy+x+6=0
2y:
[s*=+4xy+3y2:36
o
l' :"y -4y2=2
+
: 0
x'+xy=6
sxy
38.
[).#=u
41.1",
ll-f:t
|."-i+2=o
17-f-'
3s {,,_T,..\=Z
o
{*t-"='e
I x-Y:2
(r-ur+yr+3y-4=o
)
l.
, -2*r'lt
-o