Ch.10 ConicSectionsandAnalyticGeometry
10.1 TheEllipse
1 GraphEllipsesCenteredattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphtheellipseandlocatethefoci.
1) x2
81 +y2
25 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) fociat(2 14,0)and(–214,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) fociat(0,214)and(0,–214)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) fociat(5 3,0)and(–53,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(0,53)and(0,–53)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page1
2) x2
9+y2
25 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) fociat(0,4)and(0,–4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) fociat(4
,
0)and(–4
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) fociat(3 3,0)and(–33,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(0,33)and(0,–33)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page2
3) 9x2=144–16y2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) fociat(7,0)and(–7,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) fociat(0,7)and(0,–7)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) fociat(5
,
0)and(–5
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(4
,
0)and(–4
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page3
4) 16x2+9y2=144
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) fociat(0,7)and(0,–7)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) fociat(7,0)and(–7,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) fociat(5
,
0)and(–5
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(4
,
0)and(–4
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page4
5) x2
20 +y2
36 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) fociat(0,4)and(0,–4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) fociat(4
,
0)and(–4
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) fociat(0,25)and(0,–25)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(0,6)and(0,–6)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page5
6) x2
25 +y2
21 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) fociat(2
,
0)and(–2
,
0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) fociat(0,5)and(0,–5)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) fociat(21,0)and(–21,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(0,2)and(0,–2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page6
7) x2
3
2
+y2
7
2
=1
Roundtothenearesttenthifnecessary.
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) foci(0,1.4)and(0,–1.4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) foci(1.4
,
0)and(0,–1.4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) foci(0,1.4)and(0,–1.4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) foci(1.5
,
0)and(0,–1.5)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page7
2 WriteEquationsofEllipsesinStandardForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthestandardformoftheequationoftheellipseandgivethelocationofitsfoci.
1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x2
81 +y2
64 =1
fociat(–17,0)and(17,0)
B) x2
64 +y2
81 =1
fociat(–17,0)and(17,0)
C) x2
81 –y2
64 =1
fociat(–17,0)and(17,0)
D) x2
81 +y2
64 =1
fociat(–9,0)and(9,0)
2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x2
4+y2
25 =1
fociat(0,–21)and(0,21)
B) x2
25 +y2
4=1
fociat(0,–21)and(0,21)
C) x2
4+y2
25 =1
fociat(0,–5)and(0,5)
D) x2
4+y2
25 =1
fociat(0,5)and(2,0)
Page8
3)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Centerat(3,–1)
A) (x–3)2
36 +(y+1)2
25 =1
fociat(3+11,–1)and(3–11,–1)
B) (x–3)2
25 +(y+1)2
36 =1
fociat(–1+11,3)and(–1–11,3)
C) (x+1)2
25 +(y–3)2
36 =1
fociat(–11,–1)and(11,–1)
D) (x+1)2
36 +(y–3)2
25 =1
fociat(3+11,3)and(3–11,3)
Findthestandardformoftheequationoftheellipsesatisfyingthegivenconditions.
4) Foci:(–3
0),(3
,
0);vertices:(–4
0),(4
,
0)
A) x2
16 +y2
7=1B)
x2
7+y2
16 =1C)
x2
9+y2
7=1D)
x2
9+y2
16 =1
5) Foci:(0,–2),(0,2);vertices:(0,–3),(0,3)
A) x2
5+y2
9=1B)
x2
9+y2
5=1C)
x2
4+y2
5=1D)
x2
4+y2
9=1
6) Foci:(–4
0),(4
,
0);x–intercepts:–5and5
A) x2
25 +y2
9=1B)
x2
9+y2
25 =1C)
x2
16 +y2
9=1D)
x2
16 +y2
25 =1
7) Foci:(0,–2),(0,2);y–intercepts:–7and7
A) x2
45 +y2
49 =1B)
x2
49 +y2
45 =1C)
x2
4+y2
45 =1D)
x2
4+y2
49 =1
8) Majoraxishorizontalwithlength16;lengthofminoraxis=8;center(0,0)
A) x2
64 +y2
16 =1B)
x2
16 +y2
64 =1C)
x2
16 +y2
16 =1D)
x2
256 +y2
64 =1
9) Majoraxisverticalwithlength16;lengthofminoraxis=10;center(0,0)
A) x2
25 +y2
64 =1B)
x2
64 +y2
25 =1C)
x2
10 +y2
64 =1D)
x2
100 +y2
256 =1
Page9
10) Endpointsofmajoraxis:(3
,
–4)and(3
,
8);endpointsofminoraxis:(0
,
2) and(6
,
2);
A) (x–3)2
9+(y–2)2
36 =1B)
(x–3)2
9+(y–6)2
36 =1
C) (x+3)2
9+(y+2)2
36 =1D)
(x–2)2
9+(y–3)2
36 =1
11) Endpointsofmajoraxis:(–11
,
1)and(7
,
1);endpointsofminoraxis:(–2
,
–2)and(–2
,
4)
A) (x+2)2
81 +(y–1)2
9=1B)
(x–1)2
9+(y+2)2
81 =1
C) (x–2)2
81 +(y–3)2
9=0D)
(x–2)2
81 +(y–3)2
9=1
Page10
3 GraphEllipsesNotCenteredattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphtheellipse.
1) (x+2)2
9+(y–2)2
4=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page11
2) (x+1)2
4+(y–2)2
16 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page12
3) 4(x+2)2+16(y–2)2=64
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page13
4) 16(x+1)2+4(y+2)2=64
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Findthefocioftheellipsewhoseequationisgiven.
5) (x–1)2
36 +(y+1)2
9=1
A) fociat(1+33
,–1)and(1–33,–1) B) fociat(–1+33,1)and(–1–33,1)
C) fociat(–33
,–1)and(3 3,–1) D) fociat(1+33,1)and(1–33,1)
6) (x+3)2
9+(y–3)2
16 =1
A) fociat(–3,3–7)and(–3,3+7) B) fociat(3,–3–7)and(3,–3+7)
C) fociat(3,3–7)and(3,3+7) D) fociat(–2,3–7)and(–2,3+7)
Page14
7) 9(x+3)2+36(y+1)2=324
A) fociat(–3+33
,–1)and(–3–33,–1) B) fociat(–1+33,–3)and(–1–33,–3)
C) fociat(–33
,–1)and(3 3,–1) D) fociat(–3+33,–3)and(–3–33,–3)
8) 36(x–3)2+16(y+2)2=576
A) fociat(3,–2–25
)and(3,–2+25) B) fociat(–2,3–25)and(–2,3+25)
C) fociat(–3,–2–25
)and(–3,–2+25) D) fociat(4,–2–25)and(4,–2+25)
Converttheequationtothestandardformforanellipsebycompletingthesquareonxandy.
9) 16x2+25y2+32x+100y–284=0
A) (x+1)2
25 +(y+2)2
16 =1B)
(x+2)2
25 +(y+1)2
16 =1
C) (x+1)2
16 +(y+2)2
25 =1D)
(x–1)2
25 +(y–2)2
16 =1
10) 16x2+4y2–64x–24y+36=0
A) (x–2)2
4+(y–3)2
16 =1B)
(x–3)2
4+(y–2)2
16 =1
C) (x–2)2
16 +(y–3)2
4=1D)
(x+2)2
4+(y+3)2
16 =1
4 SolveAppliedProblemsInvolvingEllipses
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Thearchbeneathabridgeissemi–elliptical,aone–wayroadwaypassesunderthearch.Thewidthofthe
roadwayis32feetandtheheightofthearchoverthecenteroftheroadwayis11feet.Twotrucksplantouse
thisroad.Theyareboth8feetwide.Truck1hasanoverallheightof10feetandTruck2hasanoverallheight
of9feet.Drawaroughsketchofthesituationanddeterminewhichofthetruckscanpassunderthebridge.
A) BothTruck1andTruck2canpassunderthebridge.
B) NeitherTruck1norTruck2canpassunderthebridge.
C) Truck1canpassunderthebridge,butTruck2cannot.
D) Truck2canpassunderthebridge,butTruck1cannot.
2) Thearchbeneathabridgeissemi–elliptical,aone–wayroadwaypassesunderthearch.Thewidthofthe
roadwayis34feetandtheheightofthearchoverthecenteroftheroadwayis10feet.Twotrucksplantouse
thisroad.Theyareboth8feetwide.Truck1hasanoverallheightof9feetandTruck2hasanoverallheightof
10feet.Drawaroughsketchofthesituationanddeterminewhichofthetruckscanpassunderthebridge.
A) Truck1canpassunderthebridge,butTruck2cannot.
B) BothTruck1andTruck2canpassunderthebridge.
C) NeitherTruck1norTruck2canpassunderthebridge.
D) Truck2canpassunderthebridge,butTruck1cannot.
3) Thearchbeneathabridgeissemi–elliptical,aone–wayroadwaypassesunderthearch.Thewidthofthe
roadwayis38feetandtheheightofthearchoverthecenteroftheroadwayis13feet.Twotrucksplantouse
thisroad.Theyareboth10feetwide.Truck1hasanoverallheightof13feetandTruck2hasanoverallheight
of12feet.Drawaroughsketchofthesituationanddeterminewhichofthetruckscanpassunderthebridge.
A) Truck2canpassunderthebridge,butTruck1cannot.
B) BothTruck1andTruck2canpassunderthebridge.
C) NeitherTruck1norTruck2canpassunderthebridge.
D) Truck1canpassunderthebridge,butTruck2cannot.
Page15
5 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthesolutionsetforthesystembygraphingbothofthesystemʹsequationsinthesamerectangularcoordinate
systemandfindingpointsofintersection.
1) x2+y2=9
9x2+25y2=225
x
y
x
y
A) {(0,3),(0,–3)} B) {(3
0),(–3
,
0)} C) {(0,5),(0,–5)} D) {(5
0),(–5
,
0)}
2)
x2
16 +y2
9=1
y=3
x
y
x
y
A) {(0,3)} B) {(3
,
3)} C) {(3
,
0)} D) {(0,3),(0,–3)}
Page16
3) x2+y2=61
x+y=11
x
y
x
y
A) {(6
,
5),(5
,
6)} B) {(–6
,
5),(–5
,
6)} C) {(6
,
–5),(5
,
–6)} D) {(–6
,
–5),(–5
,
–6)}
Graphthesemi–ellipse.
4) y=–9–4x2
x
y
x
y
A)
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Page17
C)
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
–6–5–4–3–2–1 123456
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
10.2 TheHyperbola
1 LocateaHyperbolaʹsVerticesandFoci
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheverticesandlocatethefociforthehyperbolawhoseequationisgiven.
1) x2
121 –y2
81 =1
A) vertices:(–11
0),(11
,
0)
foci:(–202,0),( 202,0)
B) vertices:(–9
0),(9
,
0)
foci:(–202,0),( 202,0)
C) vertices:(0,–11),(0,11)
foci:(–202,0),( 202,0)
D) vertices:(–11
0),(11
,
0)
foci:(–9,0),(9,0)
2) y2
36 –x2
49 =1
A) vertices:(0,–6),(0,6)
foci:(0,–85),(0,85)
B) vertices:(–7
0),(7
,
0)
foci:(–85,0),(85,0)
C) vertices:(0,–6),(0,6)
foci:(–85,0),(85,0)
D) vertices:(–6
0),(6
,
0)
foci:(–7,0),(7,0)
3) 25x2–36y2=900
A) vertices:(–6
0),(6
,
0)
foci:(–61,0),(61,0)
B) vertices:(0,–6),(0,6)
foci:(0,–61),(0,61)
C) vertices:(–6
0),(6
,
0)
foci:(–11,0),(11,0)
D) vertices:(–5
0),(5
,
0)
foci:(–61,0),(61,0)
4) 49y2–64x2=3136
A) vertices:(0,–8),(0,8)
foci:(0,–113),(0,113)
B) vertices:(–8
0),(8
,
0)
foci:(–113,0),( 113,0)
C) vertices:(–7
0),(7
,
0)
foci:(–15,0),(15,0)
D) vertices:(0,–7),(0,7)
foci:(0,–113),(0,113)
Page18
5) y=±x2–11
A) vertices:(–11,0),(11,0)
foci:(–22,0),(22,0)
B) vertices:(–11
0),(11
,
0)
foci:(–11,0),(11,0)
C) vertices:(–11
0),(11
,
0)
foci:(–22,0),(22,0)
D) vertices:(0,–11),(0,11)
foci:(0,–22),(0,22)
Matchtheequationtothegraph.
6) x2
16 –y2
4=1
A)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
Page19
7) y2
16 –x2
4=1
A)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
2 WriteEquationsofHyperbolasinStandardForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthestandardformoftheequationofthehyperbolasatisfyingthegivenconditions.
1) Foci:(–7
0),(7
,
0);vertices:(–3
0),(3
,
0)
A) x2
9–y2
40 =1B)
y2
9–x2
40 =1C)
x2
9–y2
49 =1D)
y2
9–x2
49 =1
2) Foci:(0,–10),(0,10);vertices:(0,–7),(0,7)
A) y2
49 –x2
51 =1B)
x2
49 –y2
51 =1C)
x2
49 –y2
100 =1D)
y2
49 –x2
100 =1
3) Endpointsoftransverseaxis:(0,–4),(0,4);asymptote:y=2
5x
A) y2
16 –x2
100 =1B)
y2
100 –x2
16 =1C)
y2
16 –x2
25 =1D)
y2
25 –x2
4=1
4) Endpointsoftransverseaxis:(–9
,
0)
,
(9
,
0);foci:(–10
0),(–10
,
0)
A) x2
81 –y2
19 =1B)
x2
19 –y2
81 =1C)
x2
81 –y2
100 =1D)
x2
100 –y2
81 =1
Page20
5) Center:(2
,
5);Focus:(–4
,
5);Vertex:(1
,
5)
A) (x–2)2–(y–5)2
35 =1B)
(x–2)2
35 –(y–5)2=1
C) (x–5)2–(y–2)2
35 =1D)
(x–5)2
35 –(y–2)2=1
Findthestandardformoftheequationofthehyperbola.
6)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A) x2
25 –y2
4=1B)
y2
25 –x2
4=1C)
x2
4–y2
25 =1D)
y2
4–x2
25 =1
7)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A) y2
25 –x2
16 =1B)
x2
25 –y2
16 =1C)
x2
16 –y2
25 =1D)
y2
16 –x2
25 =1
Page21
8)
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A) (y–2)2
4–(x+1)2
9=1B)
(y–2)2
9–(x+1)2
4=1
C) (x+1)2
9–(y–2)2
4=1D)
(x+1)2
4–(y–2)2
9=1
Converttheequationtothestandardformforahyperbolabycompletingthesquareonxandy.
9) x2–y2–4x+2y+2=0
A) (x–2) 2–(y–1) 2=1B)(y–2) 2–(x–1) 2=1
C) (x–2) 2+(y–1) 2=1D)
(y–2) 2
4–(x–1) 2
16 =1
10) y2–9x2+2y+36x–44=0
A) (y+1)2
9–(x–2)2=1B)
(x+1)2
9–(y–2)2=1
C) (y+2)2
9–(x–4)2=1D)(x–2)2–(y+1)2
9=1
11) 9x2–16y2+18x–64y–199=0
A) (x+1)2
16 –(y+2)2
9=1B)
(x–1)2
16 –(y+2)2
9=1
C) (x+1)2
16 –(y–2)2
9=1D)
(x+1)2
9–(y+2)2
16 =1
12) 9y2–4x2+18y–8x–31=0
A) (y+1)2
4–(x+1)2
9=1B)
(y–1)2
4–(x–1)2
9=1
C) (y+1)2
9–(x+1)2
4=1D)
(x–1)2
9–(y–1)2
4=1
Page22
3 GraphHyperbolasCenteredattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useverticesandasymptotestographthehyperbola.Findtheequationsoftheasymptotes.
1) x2
4–y2
16 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Asymptotes:y=±2x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Asymptotes:y=±1
2x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Asymptotes:y=±2x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Asymptotes:y=±1
2x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page23
2) y2
9–x2
25 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Asymptotes:y=±3
5x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Asymptotes:y=±5
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Asymptotes:y=±3
5x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Asymptotes:y=±5
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page24
3) 16x2–9y2=144
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Asymptotes:y=±4
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Asymptotes:y=±3
4x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Asymptotes:y=±3
4x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Asymptotes:y=±4
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page25
4) 25y2–9x2=225
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Asymptotes:y=±3
5x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Asymptotes:y=±5
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Asymptotes:y=±3
5x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Asymptotes:y=±5
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page26
5) y=±x2–6
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Asymptotes:y=±x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Asymptotes:y= ±3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Asymptotes:y=±1
3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Asymptotes:y= ±x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
4 GraphHyperbolasNotCenteredattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthelocationofthecenter,vertices,andfociforthehyperboladescribedbytheequation.
1) (x–3)2
100 –(y+2)2
81 =1
A) Center:(3,–2);Vertices:(–7,–2)and(13,–2);Foci:(3–181,–2)and(3+181,–2)
B) Center:(–3,2);Vertices:(–13,2)and(7,2);Foci:(–3–181,2)and(–3+181,2)
C) Center:(3,–2);Vertices:(–7,2)and(13,2);Foci:(3–181,2)and(3+181,2)
D) Center:(3,–2);Vertices:(–6,–2)and(14,–2);Foci:(4+181,–1)and(–1+181,–1)
Page27
2) (y–3)2
100 –(x–4)2
36 =1
A) Center:(4,3);Vertices:(4,–7)and(4,13);Foci:(4,3–234
)and(4,3+234)
B) Center:(–4,–3);Vertices:(–4,–13)and(–4,7);Foci:(–4,–3–234
)and(–4,–3+234)
C) Center:(4,3);Vertices:(4,3–234
)and(4,3+234);Foci:(4,–7)and(4,13)
D) Center:(4,3);Vertices:(7,–6)and(5,14);Foci:(7,4–234
)and(5,4+234)
3) (x–4)2–81(y+2)2=81
A) Center:(4,–2);Vertices:(–5,–2)and(13,–2);Foci:(4–82,–2)and(4+82,–2)
B) Center:(–4,2);Vertices:(–13,2)and(5,2);Foci:(–4–82,–2)and(–4+82,–2)
C) Center:(4,–2);Vertices:(–4,–1)and(14,–1);Foci:(5–82,–1)and(5+82,–1)
D) Center:(4,–2);Vertices:(9,–2)and(–9,–2);Foci:(–82,–2)and(82,–2)
4) (y–4)2–36(x+4)2=36
A) Center:(–4,4);Vertices:(–4,–2)and(–4,10);Foci:(–4,4–37)and(–4,4+37)
B) Center:(4,–4);Vertices:(4,–10)and(4,2);Foci:(4,–4–37)and(4,–4+37)
C) Center:(–4,4);Vertices:(4,–6)and(–4,6);Foci:(–4,–37)and(–4,37)
D) Center:(–4,4);Vertices:(–3,–1)and(–3,11);Foci:(–3,5–37)and(–3,5+37)
Usethecenter,vertices,andasymptotestographthehyperbola.
5) (x–1)2
9–(y+2)2
16 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page28
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page29
6) (y+2)2
4–(x–2)2
16 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page30
7) (x+2)2–4(y+2)2=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page31
8) (y–4)2–9(x+2)2=9
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page32
9) (y–1)2–(x+2)2=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
5 SolveAppliedProblemsInvolvingHyperbolas
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) TwoLORANstationsarepositioned256 milesapartalongastraightshore.Ashiprecordsatimedifferenceof
0.00097secondsbetweentheLORANsignals.(Theradiosignalstravelat186,000milespersecond.)Wherewill
theshipreachshoreifitweretofollowthehyperbolacorrespondingtothistimedifference?Iftheshipis100
milesoffshore,whatisthepositionoftheship?
A) 38milesfromthemasterstation,(133.7
,
100) B) 90 milesfromthemasterstation,(100
,
133.7)
C) 38milesfromthemasterstation,(100
,
133.7) D) 90 milesfromthemasterstation,(133.7
,
100)
Page33
2) Tworecordingdevicesareset3600feetapart,withthedeviceatpointAtothewestofthedeviceatpointB.At
apointonalinebetweenthedevices,300feetfrompointB,asmallamountofexplosiveisdetonated.The
recordingdevicesrecordthetimethesoundreacheseachone.HowfardirectlynorthofsiteBshouldasecond
explosionbedonesothatthemeasuredtimedifferencerecordedbythedevicesisthesameasthatforthefirst
detonation?
A) 660feet B) 5886.43 feet C) 1774.82 feet D) 1554.22 feet
3) Asatellitefollowingthehyperbolicpathshowninthepictureturnsrapidlyat(0,5)andthenmovescloserand
closertotheliney=15
4xasitgetsfartherfromthetrackingstationattheorigin.Findtheequationthat
describesthepathofthesatelliteifthecenterofthehyperbolaisat(0,0).
(0,5)
y=15
4x
A) y2
25 –x2
16
9
=1B)
x2
25 –y2
(45
4)2=1C)
y2
16
9
–x2
25 =1D)
x2
(45
4)2–y2
25 =1
6 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usetherelationʹsgraphtodetermineitsdomainandrange.
1) x2
4–y2
36 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Domain:(–∞
,
–2]or[2
,
∞)
Range:(–∞,∞)
B) Domain:(–∞
,
∞)
Range:(–∞,–2)or(2,∞)
C) Domain:(–∞
,
–2]and[2
,
∞)
Range:(–∞,∞)
D) Domain:(–∞
,
∞)
Range:(–∞,∞)
Page34
2) x2
16 +y2
9=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Domain:[–4
,
4]
Range:[–3,3]
B) Domain:[–3
,
3]
Range:[–4,4]
C) Domain:(–4
,
4)
Range:(–3,3)
D) Domain:[–4
,
4]
Range:(–∞,∞)
3) y2
9–x2
16 =1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Domain:(–∞
,
∞)
Range:(–∞,–3]or[3,∞)
B) Domain:(–∞
,
∞)
Range:(–∞,–3]and[3,∞)
C) Domain:(–∞
,
–3]or[3
,
∞)
Range:(–∞,∞)
D) Domain:(–∞
,
–3]and[3
,
∞)
Range:(–∞,∞)
Page35
Findthesolutionsetforthesystembygraphingbothofthesystemʹsequationsinthesamerectangularcoordinate
systemandfindingpointsofintersection.
4) x2–y2=25
x2+y2=25
x
y
x
y
A) {(5
0),(–5
,
0)} B) {(0,5),(0,–5)} C) {(5
,
0)} D) {(0,5)}
5) 16x2+y2=16
y2–16x2=16
x
y
x
y
A) {(0,–4),(0,4)} B) {(0,–4)} C) {(0,16)} D) {(4
0),(4
,
0)}
10.3 TheParabola
1 GraphParabolaswithVerticesattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthefocusanddirectrixoftheparabolawiththegivenequation.
1) x2=24y
A) focus:(0,6)
directrix:y=–6
B) focus:(6
,
0)
directrix:y=6
C) focus:(6
,
0)
directrix:x=6
D) focus:(0,–6)
directrix:x=–6
2) x2=–16y
A) focus:(0,–4)
directrix:y=4
B) focus:(–8
,
0)
directrix:x=4
C) focus:(0,–4)
directrix:y=–4
D) focus:(0,4)
directrix:y=–4
3) y2=24x
A) focus:(6
,
0)
directrix:x=–6
B) focus:(0,6)
directrix:y=–6
C) focus:(6
,
0)
directrix:x=6
D) focus:(0,–6)
directrix:y=–6
Page36
4) y2=–32x
A) focus:(–8
,
0)
directrix:x=8
B) focus:(0,–8)
directrix:y=8
C) focus:(8
,
0)
directrix:x=–8
D) focus:(–8
,
0)
directrix:y=8
5) x=4y2
A) focus:(1
16 ,0)
directrix:x=–1
16
B) focus:(0,1
16 )
directrix:y=–1
16
C) focus:(1
4,0)
directrix:x=–1
4
D) focus:(1
16 ,0)
directrix:x=1
16
Matchtheequationtothegraph.
6) y2=10x
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page37
7) y2=–7x
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
8) x2=9y
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page38
9) x2=–5y
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page39
Graphtheparabola.
10) y2=12x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page40
11) y2=–12x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page41
12) x2=20y
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page42
13) x2=–18y
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page43
14) y2+16x=0
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
2 WriteEquationsofParabolasinStandardForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthestandardformoftheequationoftheparabolausingtheinformationgiven.
1) Focus:(25
,
0);Directrix:x=–25
A) y2=100x B) y2=–100x C) y2=25x D) x2=100y
2) Focus:(0,8);Directrix:y=–8
A) x2=32y B) x2=–32y C) y2=8x D) y2=32x
3) Vertex:(9
,
–5);Focus:(9
,
–7)
A) (x–9)2=–8(y+5) B) (x–9)2=8(y+5) C) (y–5)2=–8(x+9) D) (y–5)2=8(x+9)
Page44
4) Vertex:(6
,
–1);Focus:(2
,
–1)
A) (y+1)2=–16(x–6) B) (y+1)2=16(x–6)
C) (x+6)2=–4(y–1) D) (x+6)2=4(y–1)
5) Focus:(–6
,
–1);Directrix:x=0
A) (y+1)2=–12(x+3) B) (x+1)2=–12(y+3)
C) (y+3)2=–12(x+1) D) (x+3)2=–12(y+1)
6) Focus:(2
,
8);Directrix:y=0
A) (x–2)2=16(y–4) B) (y–2)2=16(x–4) C) (x–4)2=16(y–2) D) (y–4)2=16(x–2)
Converttheequationtothestandardformforaparabolabycompletingthesquareonxoryasappropriate.
7) y2+6y+3x–3=0
A) (y+3)2=–3(x–4) B) (y–3)2=–3(x–4) C) (y–3)2=3(x–4) D) (y+3)2=–3(x+4)
8) x2–6x+9y+0=0
A) (x–3)2=–9(y–1) B) (x+3)2=–9(y–1) C) (x+3)2=9(y–1) D) (x–3)2=–9(y+1)
3 GraphParabolaswithVerticesNotattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthevertex,focus,anddirectrixoftheparabolawiththegivenequation.
1) (y+4)2=16(x+3)
A) vertex:(–3
,
–4)
focus:(1,–4)
directrix:x=–7
B) vertex:(3
,
4)
focus:(7,4)
directrix:x=–1
C) vertex:(–4
,
–3)
focus:(0,–3)
directrix:x=–8
D) vertex:(–3
,
–4)
focus:(–7,–4)
directrix:x=1
2) (y+3)2=–16(x–1)
A) vertex:(1
,
–3)
focus:(–3,–3)
directrix:x=5
B) vertex:(–1
,
3)
focus:(–5,3)
directrix:x=3
C) vertex:(–3
,
1)
focus:(–7,1)
directrix:x=1
D) vertex:(1
,
–3)
focus:(5,–3)
directrix:x=–3
3) (x–4)2=4(y+3)
A) vertex:(4
,
–3)
focus:(4,–2)
directrix:y=–4
B) vertex:(–4
,
3)
focus:(–4,4)
directrix:y=2
C) vertex:(–3
,
4)
focus:(–3,5)
directrix:y=3
D) vertex:(4
,
–3)
focus:(4,–4)
directrix:x=–2
4) (x–2)2=–16(y–3)
A) vertex:(2
,
3)
focus:(2,–1)
directrix:y=7
B) vertex:(–2
,
–3)
focus:(–2,–7)
directrix:y=1
C) vertex:(3
,
2)
focus:(3,–2)
directrix:y=6
D) vertex:(2
,
3)
focus:(2,7)
directrix:x=–1
5) (y–3)2=16x
A) vertex:(0,3)
focus:(4,3)
directrix:x=–4
B) vertex:(0,–3)
focus:(8,–3)
directrix:x=0
C) vertex:(3
,
0)
focus:(4,3)
directrix:x=–1
D) vertex:(0,3)
focus:(–8,3)
directrix:x=–4
Page45
Matchtheequationtothegraph.
6) (y–1)2=7(x+2)
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
7) (x+1)2=6(y–2)
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page46
Graphtheparabolawiththegivenequation.
8) (y–2)2=8(x+2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page47
9) (y+1)2=–6(x–1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page48
10) (x+1)2=6(y–2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page49
11) (x–2)2=–7(y–1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
4 SolveAppliedProblemsInvolvingParabolas
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Areflectingtelescopehasaparabolicmirrorforwhichthedistancefromthevertextothefocusis34 feet.Ifthe
distanceacrossthetopofthemirroris72inches,howdeepisthemirrorinthecenter?
A) 27
34 in. B) 162
17 in. C) 9
136 in. D) 289
36 in.
2) Abridgeisbuiltintheshapeofaparabolicarch.Thebridgearchhasaspanof150feetandamaximumheight
of35feet.Findtheheightofthearchat10feetfromitscenter.
A) 34.4ft B) 0.2ft C) 2.5 ft D) 6.1 ft
Page50
3) Anexperimentalmodelforasuspensionbridgeisbuilt.Inonesection,cablerunsfromthetopofonetower
downtotheroadway,justtouchingitthere,andupagaintothetopofasecondtower.Thetowersareboth4
inchestallandstand40inchesapart.Findtheverticaldistancefromtheroadwaytothecableatapointonthe
road10inchesfromthelowestpointofthecable.
A) 1in. B) 4in. C) 1.2 in. D) 0.8 in.
4) Anexperimentalmodelforasuspensionbridgeisbuilt.Inonesection,cablerunsfromthetopofonetower
downtotheroadway,justtouchingitthere,andupagaintothetopofasecondtower.Thetowersareboth16
inchestallandstand80inchesapart.Atsomepointalongtheroadfromthelowestpointofthecable,thecable
is1.44inchesabovetheroadway.Findthedistancebetweenthatpointandthebaseofthenearesttower.
A) 28in. B) 11.8 in. C) 28.2 in. D) 12.2 in.
5) Anexperimentalmodelforasuspensionbridgeisbuilt.Inonesection,cablerunsfromthetopofonetower
downtotheroadway,justtouchingitthere,andupagaintothetopofasecondtower.Thetowersstand50
inchesapart.Atapointbetweenthetowersand15inchesalongtheroadfromthebaseofonetower,thecable
is1inchesabovetheroadway.Findtheheightofthetowers.
A) 6.25in. B) 6.75 in. C) 5.75 in. D) 8.25 in.
6) Asatellitedishisintheshapeofaparabolicsurface.Signalscomingfromasatellitestrikethesurfaceofthe
dishandarereflectedtothefocus,wherethereceiverislocated.Thesatellitedishshownhasadiameterof12
feetandadepthof3feet.Theparabolaispositionedinarectangularcoordinatesystemwithitsvertexatthe
origin.Thereceivershouldbeplacedatthefocus(0,p).Thevalueofpisgivenbytheequationa=1
4p .How
farfromthebaseofthedishshouldthereceiverbeplaced?
(6,3)
3feet
A) 3feetfromthebase B) 12 feetfromthebase
C) 1
12 feetfromthebase D) 1
3feetfromthebase
Page51
5 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Determinethedirectioninwhichtheparabolaopens,andthevertex.
1) y2–8y–x+10=0
A) Openstotheright;(–6
,
4) B) Openstotheleft;(6
,
–4)
C) Opensdownward;(4
,
6) D) Opensdownward;(–4
,
6)
2) y=x2–8x+15
A) Opensupward;(4
,
–1) B) Opensupward;(–4
,
–1)
C) Openstotheright;(–1
,
4) D) Openstotheright;(–1
,
–4)
3) x=–(y+10)2+6
A) Openstoleft;(6
,
–10) B) Openstoleft;(6
,
10)
C) Openstoleft;(–10
,
6) D) Openstoright;(–10
,
6)
Usethevertexandthedirectioninwhichtheparabolaopenstodeterminetherelationʹsdomainandrange.
4) y2–4y–x+5=0
A) Domain:(1
,
∞]
Range:(–∞,∞)
B) Domain:(–∞
,
–1)
Range:(–∞,∞)
C) Domain:(–∞
,
∞)
Range:(–∞,∞)
D) Domain:(–∞
,
∞)
Range:(–∞,–1]
5) y=x2+10x+26
A) Domain:(–∞
,
∞)
Range:[1,∞)
B) Domain:(–∞
,
∞)
Range:(1,∞)
C) Domain:(1
,
∞)
Range:(–∞,∞)
D) Domain:[1
,
∞)
Range:(–∞,∞)
6) x=–(y–7)2–6
A) Domain:(–∞
,
–6]
Range:(–∞,∞)
B) Domain:(–∞
,
–6)
Range:(–∞,∞)
C) Domain:(–∞
,
∞)
Range:(–∞,∞)
D) Domain:(–∞
,
∞)
Range:(–∞,–6]
Istherelationafunction?
7) y2–4y–x+5=0
A) Yes B) No
8) y=x2+12x+31
A) Yes B) No
9) x=–(y–8)2+3
A) Yes B) No
Page52
Findthesolutionsetforthesystembygraphingbothofthesystemʹsequationsinthesamerectangularcoordinate
systemandfindingpointsofintersection.
10) (y–6)2=x+36
y=–1
6x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) {(–36
,
6),(0,0)} B) {(36
,
–6),(0,0)} C) {(–36
0),(6
,
0)} D) {(–36
,
6)}
11) x=y2–5
x=y2–5y
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) {(–4
,
1)} B) {(1,–4)} C) {(4
,
1)} D) {(–4
,
–1)}
12) x=(y+10)2–1
(x–10)2+(y+10)2=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) {(10
,
–10)} B) {(–1,–10)}
C) {(–1,–10),(10
,
–10)} D) ∅
Page53
10.4 RotationofAxes
1 IdentifyConicsWithoutCompletingtheSquare
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifytheequationwithoutcompletingthesquare.
1) 2x2–2x+y+1=0
A) parabola B) ellipse C) hyperbola D) circle
2) 4y2–2x+2y=0
A) parabola B) circle C) hyperbola D) ellipse
3) 2x2+3y2+4x–2y=0
A) ellipse B) parabola C) circle D) hyperbola
4) 4x2+2y2+4x+2=0
A) ellipse B) parabola C) hyperbola D) circle
5) 3x2–2y2+8x+4y+4=0
A) hyperbola B) circle C) ellipse D) parabola
6) y2–3x2+7x+3y+3=0
A) hyperbola B) parabola C) ellipse D) circle
7) 5x2–6y2+2x–3y–5=0
A) hyperbola B) circle C) ellipse D) parabola
2 UseRotationofAxesFormulas
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writetheequationintermsofarotatedxʹyʹ–systemusingθ
,
theangleofrotation.Writetheequationinvolvingxʹandyʹ
instandardform.
1) x2+2xy+y2–8x+8y=0;θ=45°
A) xʹ2=–42
yʹB) 3xʹ2–42xʹyʹ+yʹ2=0
C) xʹ2=–42
yʹ2D) 2xʹ2–2xʹyʹ+2yʹ2=0
2) 5x2–6xy+5y2–8=0;θ=45°
A) xʹ2
4+yʹ2=1B)
xʹ2
4+yʹ2
4=1C)
xʹ2
4+2yʹ2=1D)xʹ2+4yʹ2=1
3) xy+16=0;θ=45°
A) yʹ2
32 –xʹ2
32 =1B)yʹ2=–32xʹC) yʹ2
32 +xʹ2
32 =1D)
xʹ2
4+yʹ2
2=1
4) 8x2+63
xy+2y2–20=0;θ=30°
A) xʹ2
20
11
–yʹ2
20 =1B)
xʹ2
20
11
+yʹ2
20 =1C)
xʹ2
1
11
–yʹ2
1=1D)
xʹ2
1
11
–yʹ2
20 =–1
Page54
Writetheappropriaterotationformulassothatinarotatedsystemtheequationhasnoxʹyʹ–term.
5) x2+2xy+y2–8x+8y=0
A) x=2
2(xʹ–yʹ);y=2
2(xʹ+yʹ)
B) x=–yʹ;y=xʹ
C) x=2+2
2xʹ– 2–2
2yʹ;y=2–2
2xʹ+2+2
2yʹ
D) x=1
2xʹ–3
2yʹ;y=3
2xʹ+1
2yʹ
6) 4x2+5xy+4y2–8x+8y=0
A) x=2
2(xʹ–yʹ);y=2
2(xʹ+yʹ)
B) x=–yʹ;y=xʹ
C) x=2+2
2xʹ– 2–2
2yʹ;y=2–2
2xʹ+2+2
2yʹ
D) x=1
2xʹ–3
2yʹ;y=3
2xʹ+1
2yʹ
7) 9x2–4xy+5y2–8x+8y=0
A) x=2–2
2xʹ– 2+2
2yʹ;y=2+2
2xʹ+2–2
2yʹ
B) x=2
2(xʹ–yʹ);y=2
2(xʹ+yʹ)
C) x=–yʹ;y=xʹ
D) x=1
2xʹ–3
2yʹ;y=3
2xʹ+1
2yʹ
8) 6x2–4xy+3y2–8x+8y=0
A) x=5xʹ–2yʹ
5;y=52xʹ+yʹ
5B) x=52xʹ–yʹ
5;y=5xʹ+2yʹ
5
C) x=3
5xʹ–4
5yʹ;y=4
5xʹ+3
5yʹD) x=7
25 xʹ–24
25 yʹ;y=24
25 xʹ+7
25 yʹ
9) 11x2–24xy+4y2+30x–40y–45=0
A) x=3
5xʹ–4
5yʹ;y=4
5xʹ+3
5yʹB) x=7
25 xʹ–24
25 yʹ;y=24
25 xʹ+7
25 yʹ
C) x=89
100 xʹ–9
20 yʹ;y=9
20 xʹ+89
100 yʹD) x=1
2xʹ–3
2yʹ;y=3
2xʹ+1
2yʹ
Page55
3 WriteEquationsofRotatedConicsinStandardForm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Rewritetheequationinarotatedxʹyʹ–systemwithoutanxʹyʹterm.Expresstheequationinvolvingxʹandyʹinthe
standardformofaconicsection.
1) 24xy–7y2+36=0
A) yʹ2
9/4 –xʹ2
4=1B)
xʹ2
4+yʹ2
9/4 =1C)
yʹ2
9–xʹ2
16 =1D)
yʹ2
4–xʹ2
9/4 =1
2) 31x2+10 3xy+21y2–144=0
A) xʹ2
4+yʹ2
9=1B)yʹ2=–42xʹC) xʹ2=–42yʹD) xʹ2
9+yʹ2
4=1
3) 4x2–4xy+y2–6x+12=0
A) yʹ+65
25
2
=65
25 xʹ–264
125 B) yʹ2=–6xʹ
C) xʹ2
6+yʹ2=1D)
xʹ2
5+yʹ2
6=1
4) x2+xy+y2–3y–6=0
A)
xʹ–2
2
2
6+
yʹ–32
2
2
18 =1B)yʹ2=–18xʹ
C) xʹ2
6–yʹ2
8=1D)
xʹ2
3+yʹ2
4=1
5) 17x2–12xy+8y2–68x+24y–12=0
A)
xʹ–25
5
2
16 +
yʹ+45
5
2
4=1B)xʹ2=–16yʹ
C) xʹ2
16 –yʹ2
4=1D)
xʹ2
4+yʹ2
16 =1
6) x2+2xy+y2+x–y–4=0
A) xʹ2=2
2yʹ+2B)yʹ2=2
2xʹC) xʹ2
2+yʹ2
4=1D)
xʹ2
2–yʹ2
4=1
Page56
Usetherotatedsystemtographtheequation.
7) 7x2+2xy+7y2=24
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
A)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
B)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
C)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
D)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
8) x2–8xy+y2+25=0
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
Page57
A)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
B)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
C) D)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
Page58
9) 3x2–23xy+y2–8x–83y=0
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
A)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
B)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
C)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
D)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
4 IdentifyConicsWithoutRotatingAxes
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifytheequationwithoutapplyingarotationofaxes.
1) x2+10xy+25y2+4x–2y–9=0
A) parabola B) ellipse C) hyperbola D) circle
2) x2–2xy–3y2+3x+4y–4=0
A) hyperbola B) ellipse C) parabola D) circle
3) x2–6xy+5y2–20=0
A) hyperbola B) parabola C) ellipse D) circle
Page59
4) 2x2+10xy+25y2–4x–2y–8=0
A) ellipse B) circle C) hyperbola D) parabola
5) 4x2+6xy+2y2+3x–4y+6=0
A) hyperbola B) ellipse C) circle D) parabola
6) 5x2–5xy+2y2+4x+2y–2=0
A) ellipse B) circle C) parabola D) hyperbola
7) 8x2+9xy+4y2+4x–4y–7=0
A) ellipse B) hyperbola C) parabola D) circle
8) 4x2–8xy+4y2–3x+6=0
A) parabola B) ellipse C) hyperbola D) circle
9) 7x2+53
xy+2y2–22=0
A) hyperbola B) parabola C) ellipse D) circle
Page60
5 Tech:RotationofAxes
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useagraphingutilitytographtheequation.
1) 2x2+43
xy+6y2+3x–y=0
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
A)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
B)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
C)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
D)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
Page61
2) x2–24xy+8y2=36
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
A)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
B)
C) D)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
Page62
3) 5x2+3xy+5y2=11
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
A)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
B)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
C)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
D)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
Page63
4) 16x2–24xy+9y2–3x–4y=0
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
A)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
B)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
C)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
D)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
5) 6x2+53xy+y2=20
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
Page64
A) B)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
C)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
D)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
6) 5x2+3xy–2y2=23
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
–8–6–4–2 2468
8
6
4
2
-2
-4
-6
-8
Page65
A) B)
C)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
D)
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
-8 -6 -4 -2 2 4 6 8
8
6
4
2
-2
-4
-6
-8
Page66
7) 3x2–4xy+3y2–2x+5y=10
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
A)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
B)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
C)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
D)
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
6 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.Expressanswersrelativetoanxʹyʹ–systeminwhichthegivenequationhasnoxʹyʹ–term.
1) 14x2+12 3xy+2y2–15=0;Findtheequationsoftheasymptotes.
A) yʹ=–5xʹandyʹ=5xʹB) yʹ= –5xʹandyʹ=5xʹ
C) yʹ=–5
5xʹandyʹ=5
5xʹD) noasymptotes
Page67
2) x2+xy+y2–3y–6=0;Findthecoordinatesoftheverticesontheminoraxis.
A) 2
2,–32
2and2
2,92
2B) 2
2,32
2and–2
2,–32
2
C) (–6,0)and(6,0) D) (0,–2)and(0,2)
3) x2+2xy+y2–8x+8y=0;Findthecoordinatesofthevertex.
A) (0,0) B) –2,0 C) (2,0) D) (–2,0)
4) 24xy–7y2+36=0;Findtheequationsoftheasymptotes.
A) yʹ=–4
3xʹandyʹ=4
3xʹB) yʹ=–3
4xʹandyʹ=3
4xʹ
C) yʹ=–2
3xʹandyʹ=2
3xʹD) yʹ=–xʹ
3andyʹ=xʹ
3
10.5 ParametricEquations
1 UsePointPlottingtoGraphPlaneCurvesDescribedbyParametricEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Parametricequationsandavaluefortheparametertaregiven.Findthecoordinatesofthepointontheplanecurve
describedbytheparametricequationscorrespondingtothegivenvalueoft.
1) x=t2+3,y=8–t3;t=2
A) (7
,
0) B) (7
,
16) C) (12
,
–5) D) (12
,
2)
2) x=6+7cost,y=8+6sint;t=π
2
A) (6
,
14) B) (13
,
8) C) 6+72
2,8+32 D) (6
,
11)
3) x=(60cos30°)t,y=4+(60sin30°)t–18t2;t=6
A) (180 3,–464) B) (360 3,346) C) (180 3,76) D) (–180 3,238)
Page68
Usepointplottingtographtheplanecurvedescribedbythegivenparametricequations.
4) x=2t,y=t+1;–2≤t≤3
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
A)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
B)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
C)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
D)
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
Page69
5) x=2t–1,y=t2+4;–4≤t≤4
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page70
6) x=t2,y=t+2;0≤t≤4
x
–20 –10 10 20
y
10
5
-5
-10
x
–20 –10 10 20
y
10
5
-5
-10
A)
x
–20 -10 10 20
y
10
5
-5
-10
x
–20 -10 10 20
y
10
5
-5
-10
B)
x
–20 -10 10 20
y
10
5
-5
-10
x
–20 -10 10 20
y
10
5
-5
-10
C)
x
–20 -10 10 20
y
10
5
-5
-10
x
–20 -10 10 20
y
10
5
-5
-10
D)
x
–20 -10 10 20
y
10
5
-5
-10
x
–20 -10 10 20
y
10
5
-5
-10
Page71
7) x=t3+1,y=t3–15;–2≤t≤2
x
–30 –15 15 30
y
40
20
-20
-40
x
–30 –15 15 30
y
40
20
-20
-40
A)
x
–30 -15 15 30
y
40
20
-20
-40
x
–30 -15 15 30
y
40
20
-20
-40
B)
x
–30 -15 15 30
y
40
20
-20
-40
x
–30 -15 15 30
y
40
20
-20
-40
C)
x
–30 -15 15 30
y
40
20
-20
-40
x
–30 -15 15 30
y
40
20
-20
-40
D)
x
–30 -15 15 30
y
40
20
-20
-40
x
–30 -15 15 30
y
40
20
-20
-40
Page72
8) x=5sint,y=5cost;0≤t≤2π
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page73
9) x=3tant,y=4sect;0≤t≤2π
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page74
10) x=t,y=3t+4;0≤t≤4
x
-5 5
y
20
10
-10
-20
x
-5 5
y
20
10
-10
-20
A)
x
-5 5
y
20
10
-10
-20
x
-5 5
y
20
10
-10
-20
B)
x
-5 5
y
20
10
-10
-20
x
-5 5
y
20
10
-10
-20
C)
x
-5 5
y
20
10
-10
-20
x
-5 5
y
20
10
-10
-20
D)
x
-5 5
y
20
10
-10
-20
x
-5 5
y
20
10
-10
-20
Page75
11) x=2t,y=|t–2|;–∞≤t≤∞
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
A)
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
B)
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
C)
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
D)
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
x
–2 24681012
y
8
6
4
2
-2
-4
-6
-8
2 EliminatetheParameter
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Eliminatetheparametert.Findarectangularequationfortheplanecurvedefinedbytheparametricequations.
1) x=3t,y=t+3;–2≤t≤3
A) y=1
3x+3;–6≤x≤9B)y= –3x+3;–∞<x<∞
C) y=1
3x–3;–∞<x<∞D) y=x2+1;–2≤x≤2
Page76
2) x=2t–1,y=t2+4;–4≤t≤4
A) y=1
4x2+1
2x+17
4;–9≤x≤7B)y=1
2x2+1;–6≤x≤4
C) y=–1
2x+30;–6≤x≤4D)y=x2+1;–2≤x≤2
3) x=6sint,y=6cost;0≤t≤2π
A) x2+y2=36;–6≤x≤6B)y
2–x2=36;–∞<x<∞
C) y=a2–x2=36;–∞<x<∞D) y=x2–9;–2≤x≤2
4) x=2tant,y=5sect;0≤t≤2π
A) y2
25 –x2
4=1;–∞<x<∞B) y2
25 +x2
4=1;–∞<x<∞
C) y=51+x2
4;–∞<x<∞D) y=x2–9;–3≤x≤3
5) x=t3+1,y=t3–1;–2≤t≤2
A) y=x–2;–7≤x≤9B)y= –x–2;–7≤x≤9
C) y=–x2;–4≤x≤4D)y=x3;–3≤x≤1
6) x=t,y=2t+3;0≤t≤4
A) y=2x2+3;0≤x≤2B)y=3x2+2;–1≤x≤2
C) y=–3x2+17;0≤x≤2D)y= –3x+17;0≤x≤2
7) x=4cost,y=4sint;0≤t≤2π
A) x2+y2=16;–4≤x≤4B)x
2+y2=4;–4≤x≤4
C) x2–y2=16;–4≤x≤4D)x
2–y2=4;–4≤x≤4
8) x=2+sect,y=5+2tant;0<t<π
2
A) (x–2)2–(y–5)2
4=1;x>3B)(x–2)2+(y–5)2
4=1;1≤x≤3
C) (y–2)2–(x–5)2
4=1;–∞<x<∞D) (x–2)2–(y–5)2=4;–∞<x<∞
9) x=et,y=e–t;–∞<t<∞
A) y=e–lnx;0<x<∞B) y=elnx;0<x<∞
C) y=ex;–∞<x<∞D) y=e–x;–∞<x<∞
Eliminatetheparameter.Writetheresultingequationinstandardform.
10) Acircle:x=5+4cost,y=2+4sint
A) (x–5)2
16 +(y–2)2
16 =1B)
(x+5)2
16 +(y+2)2
16 =1
C) (x–2)2
16 +(y–5)2
16 =1D)(x–5)2+(y–2)2=16
Page77
11) Anellipse:x=4+6cost,y=3+4sint
A) (x–4)2
36 +(y–3)2
16 =1B)
(x+4)2
36 +(y+3)2
16 =1
C) (x–3)2
16 +(y–4)2
36 =1D)(x–4)2+(y–3)2=1
12) Ahyperbola:x=2+6sect,y=4+2tant
A) (x–2)2
36 –(y–4)2
4=1B)
(x–2)2
36 +(y–4)2
4=1
C) (x+4)2
36 –(y+2)2
4=1D)
(x–4)2
4+(y–2)2
36 =1
3 FindParametricEquationsforFunctions
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findasetofparametricequationsfortheconicsectionortheline.
1) Circle:Center:(5
,
3);Radius:2
A) x=5+2cost;y=3+2sintB)x=t–5;(y–3)2+t2=4
C) x=3+2sint;y=5+2costD)x=5+sint;y=3+cost
2) Ellipse:Center:(3
,
4);Vertices:6unitsaboveandbelowthecenter;EndpointsofMinorAxis:5unitsleftand
rightofthecenter.
A) x=3+5cost,y=4+6sintB)x=3+6 cost,y=4+5sint
C) x=3–5cost,y=4–6sintD)x=5+3 cost,y=6+4sint
3) Hyperbola:Vertices:(9
,
0);Vertices:(–9
,
0);Foci:(15
,
0)and(–15
,
0)
A) x=9sect,y=12tantB)x=9 sect,y=15 tant
C) x=12sect,y=9tantD)x=81 sect,y=144tant
4) Thelinesegmentstartingat(–4
,
2)witht=0andendingat(2
,
10)witht=2
A) x=3t–4
,
y=4t+2
,
for0≤t≤2B)x=4t+2
,
y=3t–4
,
for0≤t≤2
C) x=–4t+3
,
y=2t+4
,
for0≤t≤2D)x=2t+4
,
y= –4t+3
,
for0≤t≤2
Findasetofparametricequationsfortherectangularequation.
5) y=3x–3
A) x=t;y=3t–3B)y=3t;3x=t+3C)x=t
3;y=t–1D)x=t;y=3t2–3
6) y=x4+3
A) x=t;y=t4+3B)x=t2;y=t2+3C)x=t2;y=t4+3D)x=t;y=t2+3
7) y=(x–3)2
2
A) x=t+3;y=t2
2B) x=t;y=t2–3
2
C) x=t2;y=t–3
2,x≥4D)x=(t–3)2;y=t
2,x≥0
Page78
Findtwosetsofparametricequationsforthegivenrectangularequation.
8) y=4x+7
A) x=t,y=4t+7;x=t
4,y=t+7B)x=t,y=4t+7;x=4t,y=t+7
C) x=t,y=4t+7;x=t,y=t
4+7D)x=4t,y=t+7;x=t
4,y=t+7
9) y=x2–3
A) x=t,y=t2–3;x=t,y=t–3,t≥0B)x=t,y=t2–3;x=t,y=t+3,t≥0
C) x=–t,y=t2+3;x=t,y=t–3D)x=–t,y=t2–3;x=t,y=–t+3,t≥0
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Theparametricequationsoffourcurvesaregiven.Grapheachofthem,indicatingtheorientation.
10) C1:x=7sint,y=7–7cos2t; π
2≤t≤3π
2
C2:x=lnt,y=lnt2;e–4≤t≤e3
C3:x=t2–8,y=t–3;–4≤t≤4
C4:x=t–5,y=t+2;–4≤t≤7
4 UnderstandtheAdvantagesofParametricRepresentations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Ronthrowsaballstraightupwithaninitialvelocityof70 feetpersecondfromaheightof3feet.Find
parametricequationsthatdescribethemotionoftheballasafunctionoftime.Howlongistheballintheair?
Whenistheballatitsmaximumheight?Whatisthemaximumheightoftheball?
A) x=0;y=–16t2+70t+3;
4.417sec;
2.188sec;
79.563feet
B) x=0;y=–16t2+70t+3;
8.835sec;
2.188sec;
76.563feet
C) x=0;y=–16t2+70t+3;
4.332sec;
2.188sec;
2.98feet
D) x=0;y=–16t2+70t+3;
8.663sec;
2.188sec;
603.5feet
2) Abaseballpitcherthrowsabaseballwithaninitialvelocityof138 feetpersecondatanangleof20° tothe
horizontal.Theballleavesthepitcherʹshandataheightof5feet.Findparametricequationsthatdescribethe
motionoftheballasafunctionoftime.Howlongistheballintheair?Whenistheballatitsmaximumheight?
Whatisthemaximumheightoftheball?
A) x=129.68t;y=–16t2+47.2t+5;
3.052sec;
1.475sec;
39.81feet
B) x=129.68t;y=–16t2+47.2t+5;
6.105sec;
1.475sec;
34.81feet
C) x=129.68t;y=–16t2+47.2t+5;
2.84sec;
1.475sec;
4.998feet
D) x=129.68t;y=–16t2+47.2t+5;
5.68sec;
1.475sec;
283.48feet
Page79
3) Abaseballplayerhitabaseballwithaninitialvelocityof170 feetpersecondatanangleof40°tothe
horizontal.Theballwashitataheightof4feetofftheground.Findparametricequationsthatdescribethe
motionoftheballasafunctionoftime.Howlongistheballintheair?Whenistheballatitsmaximumheight?
Whatisthehorizontaldistancetheballtraveled?
A) x=130.22t;y=–16t2+109.31t+4;
6.868sec;
3.416sec;
894.351feet
B) x=130.22t;y=–16t2+109.31t+4;
13.737sec;
3.416sec;
1788.832feet
C) x=130.22t;y=–16t2+109.31t+4;
6.795sec;
3.416sec;
884.845feet
D) x=130.22t;y=–16t2+109.31t+4;
6.868sec;
3.416sec;
1505.452feet
Page80
5 Tech:ParametricEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useagraphingutilitytoobtaintheplanecurverepresentedbythegivenparametricequations.
1) Cycloid:x=2(t–sint),y=2(1–cost),0≤t≤6π
x
10 20 30 40 50
y
8
6
4
2
x
10 20 30 40 50
y
8
6
4
2
A)
x
10 20 30 40 50
y
8
6
4
2
x
10 20 30 40 50
y
8
6
4
2
B)
x
10 20 30 40 50
y
8
6
4
2
x
10 20 30 40 50
y
8
6
4
2
C)
x
10 20 30 40 50
y
8
6
4
2
x
10 20 30 40 50
y
8
6
4
2
D)
x
10 20 30 40 50
y
8
6
4
2
x
10 20 30 40 50
y
8
6
4
2
Page81
2) WitchofAgnesi:x=3cott,y=3sin2t,0≤t<2π
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
A)
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
B)
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
C)
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
D)
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
x
-6 -4 -2 2 4 6
y
4
2
-2
-4
Page82
6 AdditionalConcepts
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Sketchtheplanecurverepresentedbythegivenparametricequations.Thenuseintervalnotationtogivetherelationʹs
domainandrange.
1) x=3t,y=t2+t+2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Domain:(–∞
,
∞);Range:[1.75
,
∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Domain:(–∞,∞);Range:–3
2x,∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Domain:(–∞
,
∞);Range:[2
,
∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Domain:(–∞
,
∞);Range:[1.75
,
∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page83
Sketchthefunctionrepresentedbythegivenparametricequations.Thenusethegraphtodetermineeachofthe
following:
a.intervals,ifany,onwhichthefunctionisincreasingandintervals,ifany,onwhichthefunctionisdecreasing.
b.thenumber,ifany,atwhichthefunctionhasamaximumandthismaximumvalue,orthenumber,ifany,atwhich
thefunctionhasaminimumandthisminimumvalue.
2) x=2t,y=t
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Increasingon:(0,∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) Increasingon:[1,∞);Minimum:(1,1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) Increasingon:(0,∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) Decreasingon:(0,∞)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page84
10.6 ConicSectionsinPolarCoordinates
1 DefineConicsinTermsofaFocusandaDirectrix
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifytheconicsectionthatthepolarequationrepresents.Describethelocationofadirectrixfromthefocuslocatedat
thepole.
1) r=6
1–2cosθ
A) hyperbola;Thedirectrixis3unit(s)totheleftofthepoleatx= –3.
B) hyperbola;Thedirectrixis3unit(s)totherightofthepoleatx=3.
C) ellipse;Thedirectrixis3unit(s)totheleftofthepoleatx= –3.
D) ellipse;Thedirectrixis3unit(s)totherightofthepoleatx=3.
2) r=9
1–3sinθ
A) hyperbola;Thedirectrixis3unit(s)belowthepoleaty= –3.
B) hyperbola;Thedirectrixis3unit(s)abovethepoleaty=3.
C) hyperbola;Thedirectrixis3unit(s)totheleftofthepoleatx= –3.
D) hyperbola;Thedirectrixis3unit(s)totherightofthepoleatx=3.
3) r=3
3+3sinθ
A) parabola;Thedirectrixis1unit(s)abovethepoleaty=1.
B) parabola;Thedirectrixis1unit(s)totherightofthepoleatx=1.
C) hyperbola;Thedirectrixis1unit(s)abovethepoleaty=1.
D) hyperbola;Thedirectrixis1unit(s)totherightofthepoleatx=1.
4) r=8
6–3sinθ
A) ellipse;Thedirectrixis8
3unit(s)belowthepoleaty=–8
3.
B) ellipse;Thedirectrixis8
3unit(s)totheleftofthepoleatx=–8
3.
C) ellipse;Thedirectrixis8
3unit(s)totherightofthepoleatx=8
3.
D) ellipse;Thedirectrixis8
3unit(s)abovethepoleaty=8
3.
5) r=4
4+2cosθ
A) ellipse;Thedirectrixis2unit(s)totherightofthepoleatx=2.
B) ellipse;Thedirectrixis2unit(s)totheleftofthepoleatx= – 2.
C) ellipse;Thedirectrixis2unit(s)belowthepoleaty= – 2.
D) ellipse;Thedirectrixis2unit(s)abovethepoleaty=2.
Page85
6) r=6
2–2cosθ
A) parabola;Thedirectrixis3unit(s)totheleftofthepoleatx= –3.
B) parabola;Thedirectrixis3unit(s)totherightofthepoleatx=3.
C) parabola;Thedirectrixis3unit(s)abovethepoleaty=3.
D) parabola;Thedirectrixis3unitsbelowthepoleaty= –3.
Page86
2 GraphthePolarEquationsofConics
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphthepolarequation.
1) r=6
3–3cosθ Identifythedirectrixandvertex.
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) directrix:2unit(s)totheleftof
thepoleatx=–2
vertex:1,π
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B) directrix:2 unit(s)totherightof
thepoleatx=2
vertex:1,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C) directrix:2unit(s)above
thepoleaty=2
vertex:1,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D) directrix:2 unit(s)below
thepoleaty=–2
vertex:1,3π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page87
2) r=6
2+2sinθ Identifythedirectrixandvertex.
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) directrix:3unit(s)above
thepoleaty=3
vertex:3
2,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B) directrix:3 unit(s)totherightof
thepoleatx=3
vertex:3
2,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C) directrix:3unit(s)totheleftof
thepoleatx=–3
vertex:–3
2,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D) directrix:3 unit(s)below
thepoleaty=–3
vertex:–3
2,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page88
3) r=4
2–cosθ Identifythedirectrixandvertices.
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) directrix:4unit(s)totheleftof
thepoleatx=–4
vertices:4
3,π ,4,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B) directrix:4 unit(s)totherightof
thepoleatx=4
vertices:–4
3,π ,4,π
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C) directrix:4unit(s)below
thepoleaty=–4
vertices:–4
3,π
2,4,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D) directrix:4 unit(s)above
thepoleaty=4
vertices:4
3,π
2,–4,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page89
4) r=8
4–sinθ Identifythedirectrixandvertices.
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) directrix:8unit(s)below
thepoleaty=–8
vertices:8
3,π
2,8
5,3π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B) directrix:8 unit(s)totherightof
thepoleatx=8
vertices:8
3,π ,8
5,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C) directrix:8unit(s)totheleftof
thepoleatx=–8
vertices:8
5,π ,8
3,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D) directrix:8 unit(s)above
thepoleaty=8
vertices:–8
5,3π
2,8
3,3π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page90
5) r=3
1+3cosθ Identifythedirectrixandvertices.
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) directrix:1unit(s)totherightof
thepoleatx=1
vertices:–3
2,π ,3
4,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B) directrix:1 unit(s)totheleftof
thepoleatx=–1
vertices:3
2,π ,–3
4,0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C) directrix:1unit(s)above
thepoleaty=1
vertices:3
4,π
2,3
2,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D) directrix:1 unit(s)below
thepoleaty=–1
vertices:–3
4,π
2,–3
2,π
2
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page91
6) r=8
4+8sinθ Identifythedirectrixandvertices.
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) directrix:1unit(s)above
thepoleaty=1
vertices:2
3,π
2,2,3π
2
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B) directrix:1 unit(s)below
thepoleaty=–1
vertices:2
3,π
2,2,3π
2
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C) directrix:1unit(s)totheleftof
thepoleatx=–1
vertices:2
3,0 ,2,π
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D) directrix:1 unit(s)totherightof
thepoleatx=1
vertices:2
3,0 ,2,π
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page92
3 Tech:ConicSectionsinPolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useagraphingutilitytographtheequation.
1) r=4
1–cosθ
–π
4
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page93
2) r=3
3+3sinθ
+π
6
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page94
4 SolveApps:ConicSectionsinPolarCoordinates
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Halleyʹscomethasanellipticalorbitwiththesunatonefocus.Itsorbitshownbelowisgivenapproximatelyby
r=10.21
1+0.874sinθ .Intheformula,rismeasuredinastronomicalunits.(Oneastronomicalunitistheaverage
distancefromEarthtothesun,approximately93millionmiles.)
FindthedistancefromHalleyʹscomettothesunatitsshortestdistancefromthesun.Roundtothenearest
hundredthofanastronomicalunitandthenearestmillionmiles.
A) 5.45astronomicalunits;507millionmiles B) 81.03 astronomicalunits;7536millionmiles
C) 11.68astronomicalunits;1086millionmiles D) 5.84 astronomicalunits;543millionmiles
Page95
2) Halleyʹscomethasanellipticalorbitwiththesunatonefocus.Itsorbitshownbelowisgivenapproximatelyby
r=10.73
1+0.859sinθ .Intheformula,rismeasuredinastronomicalunits.(Oneastronomicalunitistheaverage
distancefromEarthtothesun,approximately93millionmiles.)
FindthedistancefromHalleyʹscomettothesunatitsgreatestdistancefromthesun.Roundtothenearest
hundredthofanastronomicalunitandthenearestmillionmiles.
A) 76.1astronomicalunits;7077millionmiles B) 5.77 astronomicalunits;537millionmiles
C) 12.49astronomicalunits;1162millionmiles D) 6.25 astronomicalunits;581millionmiles
Page96
Ch.10 ConicSectionsandAnalyticGeometry
AnswerKey
10.1 TheEllipse
1 GraphEllipsesCenteredattheOrigin
2 WriteEquationsofEllipsesinStandardForm
3 GraphEllipsesNotCenteredattheOrigin
4 SolveAppliedProblemsInvolvingEllipses
5 AdditionalConcepts
10.2 TheHyperbola
1 LocateaHyperbolaʹsVerticesandFoci
Page97
2 WriteEquationsofHyperbolasinStandardForm
3 GraphHyperbolasCenteredattheOrigin
4 GraphHyperbolasNotCenteredattheOrigin
5 SolveAppliedProblemsInvolvingHyperbolas
6 AdditionalConcepts
10.3 TheParabola
1 GraphParabolaswithVerticesattheOrigin
2 WriteEquationsofParabolasinStandardForm
3 GraphParabolaswithVerticesNotattheOrigin
4 SolveAppliedProblemsInvolvingParabolas
5 AdditionalConcepts
10.4 RotationofAxes
1 IdentifyConicsWithoutCompletingtheSquare
2 UseRotationofAxesFormulas
Page99
3 WriteEquationsofRotatedConicsinStandardForm
4 IdentifyConicsWithoutRotatingAxes
5 Tech:RotationofAxes
6 AdditionalConcepts
10.5 ParametricEquations
1 UsePointPlottingtoGraphPlaneCurvesDescribedbyParametricEquations
Page100
2 EliminatetheParameter
3 FindParametricEquationsforFunctions
4 UnderstandtheAdvantagesofParametricRepresentations
Page101
5 Tech:ParametricEquations
6 AdditionalConcepts
10.6 ConicSectionsinPolarCoordinates
1 DefineConicsinTermsofaFocusandaDirectrix
2 GraphthePolarEquationsofConics
3 Tech:ConicSectionsinPolarCoordinates
4 SolveApps:ConicSectionsinPolarCoordinates
Page102