Ch.11 AnalyticGeometry
11.1 Conics
1 KnowtheNamesoftheConics
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Nametheconic.
1)
A) circle B) ellipse C) parabola D) hyperbola
2)
A) ellipse B) circle C) parabola D) hyperbola
3)
A) parabola B) circle C) ellipse D) hyperbola
Page1
4)
A) hyperbola B) circle C) ellipse D) parabola
11.2 TheParabola
1 AnalyzeParabolaswithVertexattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Matchtheequationtoitsgraph.
1) y2=9x
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Page2
2) y2=–6x
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
3) x2=8y
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Page3
4) x2=–16y
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Findanequationoftheparaboladescribed.
5) Focusat(3
,
0);directrixthelinex=–3
A) y2=12x B) y2=–12x C) y2=3x D) x2=12y
6) Focusat(0,–3);directrixtheliney=3
A) x2=–12y B) x2=12y C) y2=–3x D) y2=–12x
7) Focusat(5
,
0);vertexat(0,0)
A) y2=20x B) x2=20y C) y2=5x D) x2=5y
8) Directrixtheliney=3;vertexat(0,0)
A) y=–1
12x2B) x=3y2C) x=–1
12y2D) y=–12x2
9) Focusat(5,0);vertexat(0,0)
A) y2=20x B) y=20x2C) x2=20y D) x=20y2
10) Vertexat(0,0);axisofsymmetrythex–axis;containingthepoint(8
7)
A) y2=49
8xB)y
2=49
32xC)x
2=49
8yD)x
2=49
32y
Findanequationoftheparaboladescribedandstatethetwopointsthatdefinethelatusrectum.
11) Focusat(0,3);directrixtheliney=–3
A) x2=12y;latusrectum:(6,3)and(–6,3) B) y2=16x;latusrectum:(7,8)and(–7,8)
C) x2=12y;latusrectum:(3,6)and(–3,6) D) x2=16y;latusrectum:(8,3)and(–8,3)
Page4
Findthevertex,focus,anddirectrixoftheparabola.Graphtheequation.
12) x2=12y
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(0,0)
focus:(0,3)
directrix:y=–3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(0,0)
focus:(0,–3)
directrix:y=3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) vertex:(0,0)
focus:(3,0)
directrix:x=–3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(0,0)
focus:(–3,0)
directrix:x=3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page5
13) y2=16x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(0,0)
focus:(4,0)
directrix:x=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(0,0)
focus:(0,4)
directrix:y=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) vertex:(0,0)
focus:(0,4)
directrix:y=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(0,0)
focus:(–4,0)
directrix:x=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Graphtheequation.
14) y2=16x
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
15) y2=–6x
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
Page8
16) x2=9y
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
17) x2=–9y
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
Writeanequationfortheparabola.
18)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
(0, 0) (2, 1)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
(0, 0) (2, 1)
A) x2=4y B) x2=–4y C) y2=4x D) y2=–4x
Page10
19)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
(0, 0)
(3, 6)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
(0, 0)
(3, 6)
A) y2=12x B) x2=–12y C) x2=12y D) y2=–12x
2 AnalyzeParabolaswithVertexat(h,k)
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Matchtheequationtothegraph.
1) (y+1)2=5(x+1)
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) (y+2)2=–8(x+1)
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
3) (x+2)2=6(y–1)
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) (x+2)2=–5(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
Findanequationfortheparaboladescribed.
5) Vertexat(2
8);focusat(3
8)
A) (y–8)2=4(x–2) B) (y–8)2=–4(x–2)
C) (x–8)2=–20(y–8) D) (x–8)2=20(y–8)
6) Vertexat(5
2);focusat(5
3)
A) (x–5)2=4(y–2) B) (x–5)2=–4(y–2)
C) (y–2)2=8(x–5) D) (y–2)2=–8(x–5)
7) Vertexat(2
–8);focusat(2
–5)
A) (x–2)2=12(y+8) B) (x–2)2=–12(y+8)
C) (y–8)2=12(x+2) D) (y–8)2=–12(x+2)
8) Vertexat(9
–8);focusat(1
–8)
A) (y+8)2=–32(x–9) B) (y+8)2=32(x–9)
C) (x+9)2=28(y–8) D) (x+9)2=–28(y–8)
Findthevertex,focus,anddirectrixoftheparabolawiththegivenequation.
9) (y+3)2=20(x–1)
A) vertex:(1
–3)
focus:(6,–3)
directrix:x=–4
B) vertex:(–1
3)
focus:(4,3)
directrix:x=–6
C) vertex:(–3
1)
focus:(2,1)
directrix:x=–8
D) vertex:(1
–3)
focus:(–4,–3)
directrix:x=6
Page14
10) (y–2)2=–8(x+1)
A) vertex:(–1
2)
focus:(–3,2)
directrix:x=1
B) vertex:(1
–2)
focus:(–1,–2)
directrix:x=3
C) vertex:(2
–1)
focus:(0,–1)
directrix:x=4
D) vertex:(–1
2)
focus:(1,2)
directrix:x=–3
11) (x+4)2=4(y–1)
A) vertex:(–4
1)
focus:(–4,2)
directrix:y=0
B) vertex:(4
–1)
focus:(4,0)
directrix:y=–2
C) vertex:(1
–4)
focus:(1,–3)
directrix:y=–5
D) vertex:(–4
1)
focus:(–4,0)
directrix:x=2
12) (x–4)2=–4(y–2)
A) vertex:(4
2)
focus:(4,1)
directrix:y=3
B) vertex:(–4
–2)
focus:(–4,–3)
directrix:y=–1
C) vertex:(2
4)
focus:(2,3)
directrix:y=5
D) vertex:(4
2)
focus:(4,3)
directrix:x=1
Findthevertex,focus,anddirectrixoftheparabola.Graphtheequation.
13) (y+2)2=–8(x–3)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(3
–2)
focus:(1,–2)
directrix:x=5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(2
–3)
focus:(0,–3)
directrix:x=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) vertex:(3
–2)
focus:(3,–4)
directrix:y=0
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(–3
2)
focus:(–3,0)
directrix:y=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
14) (x–1)2=–(y–2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(1
2)
focus:(1,1.75)
directrix:y=2.25
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(–1
–2)
focus:(–1,–2.25)
directrix:y=–1.75
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) vertex:(1
2)
focus:(0.75,2)
directrix:x=1.25
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(–1
–2)
focus:(–1.25,–2)
directrix:x=–0.75
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
15) x2–6x=8y–41
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(3
4)
focus:(3,6)
directrix:y=2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(3
4)
focus:(3,2)
directrix:y=6
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) vertex:(3
4)
focus:(5,4)
directrix:x=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(3
4)
focus:(1,4)
directrix:x=5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
16) y2+14y=12x–1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(–4
–7)
focus:(–1,–7)
directrix:x=–7
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(–4
–7)
focus:(–7,–7)
directrix:x=–1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page18
C) vertex:(–4
–7)
focus:(–4,–4)
directrix:y=–10
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(–4
–7)
focus:(–4,–10)
directrix:y=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Graphtheequation.
17) (y–2)2=5(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
18) (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
19) (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
20) (x+2)2=–5(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
3 SolveAppliedProblemsInvolvingParabolas
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Areflectingtelescopecontainsamirrorshapedlikeaparaboloidofrevolution.Ifthemirroris22 inches
acrossatitsopeningandis3feetdeep,wherewillthelightbeconcentrated?
A) 0.8in.fromthevertex B) 10.1 in.fromthevertex
C) 0.4in.fromthevertex D) 0.2 in.fromthevertex
2) Asearchlightisshapedlikeaparaboloidofrevolution.Ifthelightsourceislocated3feetfromthebase
alongtheaxisofsymmetryandtheopeningis8feetacross,howdeepshouldthesearchlightbe?
A) 1.3ft B) 0.6ft C) 5.3 ft D) 4ft
3) Abridgeisbuiltintheshapeofaparabolicarch.Thebridgearchhasaspanof166feetandamaximum
heightof35feet.Findtheheightofthearchat15feetfromitscenter.
A) 33.9ft B) 0.3ft C) 4.6 ft D) 15.2 ft
4) Areflectingtelescopehasamirrorshapedlikeaparaboloidofrevolution.Ifthedistanceofthevertexto
thefocusis29feetandthedistanceacrossthetopofthemirroris74inches,howdeepisthemirrorinthe
center?
A) 1369
1392
in. B) 1369
116
in. C) 1369
16704
in. D) 841
148
in.
5) Anexperimentalmodelforasuspensionbridgeisbuiltintheshapeofaparabolicarch.Inonesection,
cablerunsfromthetopofonetowerdowntotheroadway,justtouchingitthere,andupagaintothetop
ofasecondtower.Thetowersareboth6.25inchestallandstand50inchesapart.Findtheverticaldistance
fromtheroadwaytothecableatapointontheroad12.5inchesfromthelowestpointofthecable.
A) 1.56in. B) 6.25 in. C) 1.76 in. D) 1.36 in.
6) Anexperimentalmodelforasuspensionbridgeisbuiltintheshapeofaparabolicarch.Inonesection,
cablerunsfromthetopofonetowerdowntotheroadway,justtouchingitthere,andupagaintothetop
ofasecondtower.Thetowersareboth4inchestallandstand40inchesapart.Atsomepointalongthe
roadfromthelowestpointofthecable,thecableis0.36inchesabovetheroadway.Findthedistance
betweenthatpointandthebaseofthenearesttower.
A) 14in. B) 5.8in. C) 14.2 in. D) 6.2in.
7) Anexperimentalmodelforasuspensionbridgeisbuiltintheshapeofaparabolicarch.Inonesection,
cablerunsfromthetopofonetowerdowntotheroadway,justtouchingitthere,andupagaintothetop
ofasecondtower.Thetowersstand80inchesapart.Atapointbetweenthetowersand28inchesalongthe
roadfromthebaseofonetower,thecableis1.44inchesabovetheroadway.Findtheheightofthetowers.
A) 16in. B) 16.5 in. C) 15.5 in. D) 18in.
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
8) Asatellitedishisshapedlikeaparaboloidofrevolution.Thesignalsthatemanatefromasatellitestrike
thesurfaceofthedishandarereflectedtoasinglepoint,wherethereceiverislocated.Ifthedishis8feet
acrossatitsopeningandis2feetdeepatitscenter,atwhatpositionshouldthereceiverbeplaced?
9) Asealed–beamheadlightisintheshapeofaparaboloidofrevolution.Thebulb,whichisplacedatthe
focus,is3centimetersfromthevertex.Ifthedepthistobe6centimeters,whatisthediameterofthe
headlightatitsopening?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
10) Aspotlighthasaparaboliccrosssectionthatis6ftwideattheopeningand2.5ftdeepatthevertex.How
farfromthevertexisthefocus?Roundanswertotwodecimalplaces.
A) 0.90ft B) 0.52ft C) 0.21ft D) 0.26ft
Page23
11.3 TheEllipse
1 AnalyzeEllipseswithCenterattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Matchthegraphtoitsequation.
1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y2
36
+x2
4
=1B)
y2
36
–x2
4
=1C)
x2
4
–y2
36
=1D)
x2
36
+y2
4
=1
2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y2
16
+x2
25
=1B)
y2
16
–x2
25
=1C)
–y2
16
+x2
25
=1D)
x2
16
+y2
25
=1
Findthecenter,foci,andverticesoftheellipse.
3) x2
9
+y2
4
=1
A) centerat(0,0)
fociat(–5,0)and(5,0)
verticesat(–3,0),(3,0)
B) centerat(0,0)
fociat(0,–5)and(0,5)
verticesat(0,–3),(0,3)
C) centerat(0,0)
fociat(–3,0)and(3,0)
verticesat(–9,0),(9,0)
D) centerat(0,0)
fociat(0,–2)and(0,2)
verticesat(0,–4),(0,4)
Page24
4) x2
49
+y2
81
=1
A) centerat(0,0)
fociat(0,–42
)and(0,42)
verticesat(0,–9),(0,9)
B) centerat(0,0)
fociat(–42,0)and(4 2,0)
verticesat(–9,0),(9,0)
C) centerat(0,0)
fociat(0,–9)and(0,9)
verticesat(0,–81),(0,81)
D) centerat(0,0)
fociat(0,9)and(7,0)
verticesat(0,81),(49,0)
5) 25x2+64y2=1600
A) centerat(0,0)
fociat(–39,0)and(39,0)
verticesat(–8,0),(8,0)
B) centerat(0,0)
fociat(0,–39)and(0,39)
verticesat(0,–8),(0,8)
C) centerat(0,0)
fociat(–8,0)and(8,0)
verticesat(–64,0),(64,0)
D) centerat(0,0)
fociat(0,–5)and(0,5)
verticesat(0,–25),(0,25)
6) 64x2+16y2=1024
A) centerat(0,0)
fociat(0,–43
)and(0,43)
verticesat(0,–8),(0,8)
B) centerat(0,0)
fociat(–43,0)and(4 3,0)
verticesat(–8,0),(8,0)
C) centerat(0,0)
fociat(0,–8)and(0,8)
verticesat(0,–64),(0,64)
D) centerat(0,0)
fociat(0,8)and(4,0)
verticesat(0,64)and(16,0)
Findanequationfortheellipsedescribed.
7) Centerat(0,0);focusat(3
,
0);vertexat(5
,
0)
A) x2
25
+y2
16
=1B)
x2
16
+y2
25
=1C)
x2
9
+y2
16
=1D)
x2
9
+y2
25
=1
8) Centerat(0,0);focusat(–2
,
0);vertexat(5
,
0)
A) x2
25
+y2
21
=1B)
x2
21
+y2
25
=1C)
x2
4
+y2
21
=1D)
x2
4
+y2
25
=1
9) Centerat(0,0);focusat(–2
,
0);vertexat(8
,
0)
A) x2
64
+y2
60
=1B)
x2
60
+y2
64
=1C)
x2
4
+y2
60
=1D)
x2
4
+y2
64
=1
10) Centerat(0,0);focusat(0,6);vertexat(0,8)
A) x2
28
+y2
64
=1B)
x2
64
+y2
28
=1C)
x2
36
+y2
28
=1D)
x2
36
+y2
64
=1
11) Centerat(0,0);focusat(0,–4);vertexat(0,8)
A) x2
48
+y2
64
=1B)
x2
64
+y2
48
=1C)
x2
16
+y2
48
=1D)
x2
16
+y2
64
=1
12) Centerat(0,0);focusat(0,2);vertexat(0,–8)
A) x2
60
+y2
64
=1B)
x2
64
+y2
60
=1C)
x2
4
+y2
60
=1D)
x2
4
+y2
64
=1
13) Focusat(–4
,
0);verticesat(±6
,
0)
A) x2
36
+y2
20
=1B)
x2
20
+y2
36
=1C)
x2
16
+y2
20
=1D)
x2
16
+y2
36
=1
14) Focusat(0,–2);verticesat(0,±3)
A) x2
5
+y2
9
=1B)
x2
9
+y2
5
=1C)
x2
4
+y2
5
=1D)
x2
4
+y2
9
=1
15) Fociat(±2
,
0);x–interceptsare±6
A) x2
36
+y2
32
=1B)x2
32
+y2
36
=1C)
x2
4
+y2
32
=1D)
x2
4
+y2
36
=1
16) Fociat(0,±2);y–interceptsare±7
A) x2
45
+y2
49
=1B)x2
49
+y2
45
=1C)
x2
4
+y2
45
=1D)
x2
4
+y2
49
=1
17) Center(0,0);majoraxishorizontalwithlength18;lengthofminoraxisis4
A) x2
81
+y2
4
=1B)
x2
4
+y2
81
=1C)
x2
18
+y2
4
=1D)
x2
324
+y2
16
=1
18) Center(0,0);majoraxisverticalwithlength14;lengthofminoraxisis10
A) x2
25
+y2
49
=1B)
x2
49
+y2
25
=1C)
x2
10
+y2
49
=1D)
x2
100
+y2
196
=1
Graphtheellipseandlocatethefoci.
19) 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) 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(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
D) 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
20) x2
9
+y2
16
=1
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
21) 4x2+9y2=36
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) 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
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(13,0)and(–13,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(2 3,0)and(–23,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page29
22) 9x2+4y2=36
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) 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
B) 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
C) fociat(13,0)and(–13,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) fociat(2 3,0)and(–23,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page30
2 AnalyzeEllipseswithCenterat(h,k)
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writeanequationforthegraph.
1)
x
-5 5
y
5
-5
(1, –2)
x
-5 5
y
5
-5
(1, –2)
A) (x–1)2
16
+(y+2)2
9
=1B)
(x+2)2
16
+(y–1)2
9
=1
C) (x+1)2
16
+(y–2)2
9
=1D)
(x–1)2
9
+(y+2)2
16
=1
Findthecenter,foci,andverticesoftheellipse.
2) (x–3)2
16
+(y+2)2
9
=1
A) centerat(3
–2)
fociat(3+7,–2),(3–7,–2)
verticesat(–1,–2),(7,–2)
B) centerat(–2
3)
fociat(–2+7,3),(–2–7,3)
verticesat(–1,–2),(7,–2)
C) centerat(3
–2)
fociat(–7,–2),(7,–2)
verticesat(4,–2),(–4,–2)
D) centerat(3
–2)
fociat(3+7,3),(3–7,3)
verticesat(4,–2),(–4,–2)
3) 36(x–2)2+16(y+1)2=576
A) centerat(2
–1)
fociat(2,–1–25
),(2,–1+25)
verticesat(2,5),(2,–7)
B) centerat(–1
2)
fociat(–1,2–25
),(–1,2+25)
verticesat(–1,5),(–1,–7)
C) centerat(–2
–1)
fociat(–2,–1–25
),(–2,–1+25)
verticesat(–2,5),(–2,–7)
D) centerat(3
–1)
fociat(3,–1–25
),(3,–1+25)
verticesat(3,5),(3,–7)
Page31
4) 2x2+3y2–20x+18y+71=0
A) (x–5)2
3
+(y+3)2
2
=1
center:(5,–3);foci:(6,–3),(4,–3);vertices:(6.7,–3),(3.3,–3)
B) (x–5)2
2
+(y+3)2
3
=1
center:(5,–3);foci:(6,–3),(4,–3);vertices:(6.7,–3),(3.3,–3)
C) (x–5)2
3
+(y+3)2
2
=1
center:(–5,3);foci:(–4,3),(–6,3);vertices:(–6.7,3),(–3.3,3)
D) (x–5)2
2
+(y+3)2
3
=1
center:(–5,3);foci:(–4,3),(–6,3);vertices:(–6.7,3),(–3.3,3)
5) 49x2+y2–294x+392=0
A) (x–3)2+y2
49
=1
center:(3,0);foci:(3,43
),(3,–43);vertices:(3,7),(3,–7)
B) x2
49
+(y–3)2=1
center:(3,0);foci:(3,43
),(3,–43);vertices:(3,7),(3,–7)
C) (x–7)2+y2
9
=1
center:(7,0);foci:(7,22
),(7,–22);vertices:(7,3),(7,–3)
D) x2
9
+(y–7)2=1
center:(7,0);foci:(7,22
),(7,–43);vertices:(7,3),
(7,–3)
Graphtheequation.
6) (x+2)2
16
+(y–1)2
9
=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
7) (x–2)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
Page34
8) 4(x+2)2+9(y+1)2=36
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
9) 16(x+2)2+9(y+1)2=144
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
Findanequationfortheellipsedescribed.
10) Centerat(4
3);focusat(8
3);vertexat(10
3)
A) (x–4)2
36
+(y–3)2
20
=1B)
(x+4)2
36
+(y+3)2
20
=1
C) (x–4)2
81
+(y+3)2
8
=2D)
(x+4)2
16
–(y–3)2
16
=1
11) Verticesat(–5
2)and(15
2);focusat(13
2)
A) (x–5)2
100
+(y–2)2
36
=1B)
(x–2)2
81
+(y–5)2
35
=1
C) (x+5)2
64
+(y+2)2
36
=1D)
(x–5)2
144
–(y+2)2
44
=1
Page36
Findanequationfortheellipsedescribed.Graphtheequation.
12) Fociat(–2
–1)and(–2
–7);lengthofmajoraxisis10
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
(y+4)2
25
+(x+2)2
16
=1
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x+4)2
25
+(y–2)2
16
=1
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(y+4)2
25
+(x–2)2
16
=1
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x+4)2
16
+(y–2)2
25
=1
Page37
13) Fociat(–6
–3)and(0
–3);lengthofmajoraxisis10
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
(x+3)2
25
+(y+3)2
16
=1
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(y–3)2
25
+(x+3)2
16
=1
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x+3)2
25
+(x–3)2
16
=1
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x–3)2
25
+(y–3)2
16
=1
Page38
14) Verticesat(5,–4)and(5,8);lengthofminoraxisis6
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
(x–5)2
9
+(y–2)2
36
=1
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x–5)2
36
+(y–2)2
9
=1
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x+5)2
36
+(y+2)2
9
=1
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
(x–5)2
9
–(y–2)2
36
=1
15) Fociat(–2
2)and(–8
2);vertexat(–9
2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (x+5)2
16
+(y–2)2
7
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) (x+5)2
7
+(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
C) (x–2)2
16
+(y+5)2
7
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) (x–2)2
7
+(y+5)2
16
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
16) Centerat(–2
4);focusat(–8
4);containsthepoint(–10
4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (x+2)2
64
+(y–4)2
28
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) (x+2)2
28
+(y–4)2
64
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) (x+4)2
64
+(y–2)2
28
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) (x+4)2
28
+(y–2)2
64
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
3 SolveAppliedProblemsInvolvingEllipses
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphthefunction.
1) y=–9–4x2
x
y
x
y
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Solvetheproblem.
2) Abridgeisbuiltintheshapeofasemiellipticalarch.Ithasaspanof106 feet.Theheightofthearch29 feet
fromthecenteristobe8feet.Findtheheightofthearchatitscenter.
A) 9.56ft B) 8.32 ft C) 29.34 ft D) 14.62 ft
3) Anarchforabridgeoverahighwayisintheformofasemiellipse.Thetopofthearchis35 feetabove
ground(themajoraxis).Whatshouldthespanofthebridgebe(thelengthofitsminoraxis)iftheheight
29feetfromthecenteristobe15feetaboveground?
A) 64.19ft B) 32.1 ft C) 135.33 ft D) 53.58 ft
Page42
4) Theorbitofaplanetaroundasunisanellipsewiththesunatonefocus.Theaphelionofaplanetisits
greatestdistancefromthesun,itsperihelionisitsshortestdistance,anditsmeandistanceisthelengthof
thesemimajoraxisoftheellipticalorbit.Ifaplanethasaperihelionof546.5millionmilesandamean
distanceof549millionmiles,writeanequationfortheorbitoftheplanetaroundthesun.
A) x2
5492
+y2
548.9942
=1B)
x2
549.0062
+y2
5492
=1
C) x2
5492
+y2
2.52
=1D)
x2
5492
+y2
546.52
=1
5) Anarchintheformofasemiellipseis52ftwideatthebaseandhasaheightof20ft.Howwideisthearch
ataheightof12ftabovethebase?
A) 41.6ft B) 20.8ft C) 35.5ft D) 17.7ft
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
6) Ahall130feetinlengthwasdesignedasawhisperinggallery.Iftheceilingis25feethighatthecenter,
howfarfromthecenterarethefocilocated?
7) Aracetrackisintheshapeofanellipse80feetlongand60feetwide.Whatisthewidth32feetfromthe
center?
Page43
11.4 TheHyperbola
1 AnalyzeHyperbolaswithCenterattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Matchtheequationtothegraph.
1) x2
16
–y2
9
=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) y2
4
–x2
16
=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
Findanequationforthehyperboladescribed.
3) Verticesat(0,±10);asymptotesaty=±5
2x
A) y2
100
–x2
16
=1B)
y2
16
–x2
100
=1C)
y2
100
–x2
4
=1D)
y2
4
–x2
25
=1
4) Verticesat(±7
,
0);fociat(±9
,
0)
A) x2
49
–y2
32
=1B)
x2
32
–y2
49
=1C)
x2
49
–y2
81
=1D)
x2
81
–y2
49
=1
Findanequationforthehyperboladescribed.Graphtheequation.
5) Centerat(0,0);focusat(4 5,0);vertexat(8,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x2
64
–y2
16
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) x2
16
–y2
64
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) y2
16
–x2
64
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) y2
64
–x2
16
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
6) Centerat(0,0);vertexat(0,5);focusat(0,89)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y2
25
–x2
64
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) y2
64
–x2
25
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) x2
64
–y2
25
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) x2
25
–y2
64
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Findthecenter,transverseaxis,vertices,foci,andasymptotesofthehyperbola.
7) x2
144
–y2
100
=1
A) centerat(0,0)
transverseaxisisx–axis
verticesat(–12,0)and(12,0)
fociat(–261
,0)and(2 61,0)
asymptotesofy=–5
6
andy=5
6
B) centerat(0,0)
transverseaxisisx–axis
verticesat(–10,0)and(10,0)
fociat(–261
,0)and(2 61,0)
asymptotesofy=–5
6
andy=5
6
C) centerat(0,0)
transverseaxisisy–axis
verticesat(0,–12)and(0,12)
fociat(–261,0)and(2 61,0)
asymptotesofy=–5
6
andy=5
6
D) centerat(0,0)
transverseaxisisx–axis
verticesat(–12,0)and(12,0)
fociat(–10,0)and(10,0)
asymptotesofy=–5
6
andy=5
6
8) 16y2–144x2=2304
A) centerat(0,0)
transverseaxisisy–axis
verticesat(0,–12)and(0,12)
fociat(0,–410
)and(0,410)
asymptotesofy=–3andy=3
B) centerat(0,0)
transverseaxisisx–axis
vertices:(–4,0),(4,0)
foci:(–410,0),(4 10,0)
asymptotesofy=–3andy=3
C) centerat(0,0)
transverseaxisisy–axis
vertices:(0,–12),(0,12)
foci:(–410
,0),(4 10,0)
asymptotesofy=–3andy=3
D) centerat(0,0)
transverseaxisisx–axis
vertices:(–12,0),(12,0)
foci:(–4,0),(4,0)
asymptotesofy=–3andy=3
Writeanequationforthehyperbola.
9)
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
16
=1B)
y2
25
–x2
16
=1C)
x2
16
–y2
25
=1D)
y2
16
–x2
25
=1
10)
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
4
–x2
25
=1B)
x2
4
–y2
25
=1C)
x2
25
–y2
4
=1D)
y2
25
–x2
4
=1
Graphthehyperbola.
11) x2
9
–y2
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
Page50
12) y2
9
–x2
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
13) 16x2–4y2=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
14) 9y2–4x2=36
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
15) 16x2=4y2+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
16) 36y2=4x2+144
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 FindtheAsymptotesofaHyperbola
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheasymptotesofthehyperbola.
1) x2
9
–y2
16
=1
A) y=4
3xandy=–4
3xB)y=3
4xandy=–3
4x
C) y=16
9xandy=–16
9xD)y=9
16xandy=–9
16x
2) y2–x2=4
A) y=xandy=–xB)y=2xandy= – 2x
C) y=1
2xandy=–1
2xD)y=1
4xandy=–1
4x
3) (x–2)2
25
–(y+1)2
9
=1
A) y+1=3
5(x–2)andy+1=–3
5(x–2) B) y+1=5
3(x–2)andy+1=–5
3(x–2)
C) y=3
5(x–2)andy=–3
5(x–2) D) y–2=3
5(x+1)andy–2=–3
5(x+1)
4) x2–y2+4x–6y–30=0
A) y+3=(x+2)andy+3=–(x+2) B) y+3=1
5(x+2)andy+3=–1
5(x+2)
C) y–2=(x–3)andy–2=–(x–3) D) y+2=(x+3)andy+2=–(x+3)
3 AnalyzeHyperbolaswithCenterat(h,k)
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findanequationforthehyperboladescribed.
1) Vertices(1
2,–3)and(–9
2,–3);asymptotesy+3=±6
5(x+2)
A) 4(x+2)2
25
–(y+3)2
9
=1B)
(x+2)2
9
–4(y+3)2
25
=1
C) (y+3)2
9
–4(x+2)2
25
=1D)
4(x–2)2
25
–(y–3)2
9
=1
2) centerat(4
1);focusat(0
1);vertexat(3
1)
A) (x–4)2–(y–1)2
15
=1B)
(x–4)2
15
–(y–1)2=1
C) (x–1)2–(y–4)2
15
=1D)
(x–1)2
15
–(y–4)2=1
3) Verticesat(0,±10);asymptotesaty=±5
2x
A) y2
100
–x2
16
=1B)
y2
16
–x2
100
=1C)y2
100
–x2
4
=1D)
y2
4
–x2
25
=1
4) Verticesat(±10
,
0);fociat(±11
,
0)
A) x2
100
–y2
21
=1B)
x2
21
–y2
100
=1C)
x2
100
–y2
121
=1D)
x2
121
–y2
100
=1
Findthecenter,transverseaxis,vertices,foci,andasymptotesofthehyperbola.
5) (x+1)2
4
–(y+4)2
36
=1
A) centerat(–1
–4)
transverseaxisisparalleltox–axis
verticesat(–3,–4)and(1,–4)
fociat(–1–210
,–4)and(–1+210,–4)
asymptotesofy+4=–3(x+1)andy+4=3(x+1)
B) centerat(–4
–1)
transverseaxisisparalleltox–axis
verticesat(–6,–1)and(–2,–1)
fociat(–4–210,–1)and(–4+210,–1)
asymptotesofy+1=–3(x+4)andy+1=3(x+4)
C) centerat(–1
–4)
transverseaxisisparalleltoy–axis
verticesat(–1,–6)and(–1,–2)
fociat(–1,–4–210)and(–1,–4+210)
asymptotesofy–4=–1
3(x–1)andy–4=1
3(x–1)
D) centerat(–1
–4)
transverseaxisisparalleltox–axis
verticesat(–7,–4)and(5,–4)
fociat(–1–210,–4)and(–1+210,–4)
asymptotesofy+4=–1
3(x+1)andy+4=1
3(x+1)
6) (x–2)2–25(y+2)2=25
A) centerat(2
–2)
transverseaxisisparalleltox–axis
verticesat(–3,–2)and(7,–2)
fociat(2–26,–2)and(2+26,–2)
asymptotesofy+2=–1
5(x–2)andy+2=1
5(x–2)
B) centerat(–2
2)
transverseaxisisparalleltox–axis
verticesat(–7,2)and(3,2)
fociat(–2–26,2)and(–2+26,2)
asymptotesofy–2=–1
5(x+2)andy–2=1
5(x+2)
C) centerat(2
–2)
transverseaxisisparalleltoy–axis
verticesat(2,–7)and(2,3),
fociat(2,–2–26)and(2,–2+26),
asymptotesofy–2=–5(x+2)andy–2=5(x+2)
D) centerat(2
–2)
transverseaxisisparalleltox–axis
verticesat(1,–2)and(3,–2)
fociat(2–26,–2)and(2+26,–2)
asymptotesofy+2=–5(x–2)andy+2=5(x–2)
7) x2–16y2+4x–32y–28=0
A) centerat(–2
–1)
transverseaxisisparalleltox–axis
verticesat(–6,–1)and(2,–1)
fociat(–2–17,–1)and(–2+17,–1)
asymptotesofy+1=–1
4(x+2)andy+1=1
4(x+2)
B) centerat(–1
–2)
transverseaxisisparalleltox–axis
verticesat(–5,–2)and(3,–2)
fociat(–1–17,–2)and(–1+17,–2)
asymptotesofy+2=–1
4(x+1)andy+2=1
4(x+1)
C) centerat(–2
–1)
transverseaxisisparalleltoy–axis
verticesat(–2,–5)and(–2,3)
fociat(–2,–1–17)and(–2,–1+17)
asymptotesofy–1=–4(x–2)andy–1=4(x–2)
D) centerat(–2
–1)
transverseaxisisparalleltox–axis
verticesat(–3,–1)and(–1,–1)
fociat(–2–17,–1)and(–2+17,–1)
asymptotesofy+1=–4(x+2)andy+1=4(x+2)
Graphthehyperbola.
8) (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
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
9) (y–2)2
9
–(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
10) (x+2)2–4(y+1)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
11) (y–3)2–4(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
4 SolveAppliedProblemsInvolvingHyperbolas
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Tworecordingdevicesareset2000feetapart,withthedeviceatpointAtothewestofthedeviceatpoin
t
B.Atapointonalinebetweenthedevices,200feetfrompointB,asmallamountofexplosiveis
detonated.Therecordingdevicesrecordthetimethesoundreacheseachone.Howfardirectlynorthof
siteBshouldasecondexplosionbedonesothatthemeasuredtimedifferencerecordedbythedevicesis
thesameasthatforthefirstdetonation?
A) 450ft B) 2939.39 ft C) 979.8 ft D) 960.47 ft
2) Theroofofabuildingisintheshapeofthehyperbolay2–x2=19,wherexandyareinmeters.Referto
thefigureanddeterminetheheighthoftheoutsidewalls.
a=b=6m
A) 7.4mB)55mC)17mD)13m
3) Theroofofabuildingisintheshapeofthehyperbolay2–x2=6,wherexandyareinmeters.Determine
thedistance,w,theoutsidewallsareapart,iftheheightofeachwallis5m.
A) 8.7mB)19m C) 4.35 m D) 5.6m
4) Acometfollowsthehyperbolicpathdescribedbyx2
21
–y2
6
=1,wherexandyareinmillions.Ifthesunis
thefocusofthepath,howclosetothesunisthevertexofthepath?
A) 0.6million B) 5.2million C) 4.6 million D) 27million
Page63
5) Asatellitefollowingthehyperbolicpathshowninthepictureturnsrapidlyat(0,3)andthenmovescloser
andclosertotheliney=12
5xasitgetsfartherfromthetrackingstationattheorigin.Findtheequationthat
describesthepathoftherocketifthecenterofthehyperbolaisat(0,0).
(0,3)
y=12
5x
A) y2
9
–x2
25
16
=1B)
x2
9
–y2
48
5
2
=1C)
y2
25
16
–x2
9
=1D)
x2
48
5
2
–y2
9
=1
11.5 RotationofAxes;GeneralFormofaConic
1 IdentifyaConic
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifytheequationwithoutcompletingthesquare.
1) 2x2–4x+y+1=0
A) parabola B) ellipse C) hyperbola D) notaconic
2) 4y2–3x–4y=0
A) parabola B) ellipse C) hyperbola D) notaconic
3) 4x2+2y2+2x–2y=0
A) ellipse B) parabola C) hyperbola D) notaconic
4) 2x2+4y2+6x+4=0
A) ellipse B) parabola C) hyperbola D) notaconic
5) 2x2–4y2+5x+2y+4=0
A) hyperbola B) parabola C) ellipse D) notaconic
6) y2–2x2+6x+3y+1=0
A) hyperbola B) parabola C) ellipse D) notaconic
Page64
2 UseaRotationofAxestoTransformEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Determinetheappropriaterotationformulastousesothatthenewequationcontainsnoxy–term.
1) x2+2xy+y2–8x+8y=0
A) x=2
2(xʹ–yʹ)andy=2
2(xʹ+yʹ)
B) x=–yʹandy=xʹ
C) x=2+2
2xʹ– 2–2
2yʹandy=2–2
2xʹ+2+2
2yʹ
D) x=1
2xʹ–3
2yʹandy=3
2xʹ+1
2yʹ
2) 4x2+4xy+4y2–8x+8y=0
A) x=2
2(xʹ–yʹ)andy=2
2(xʹ+yʹ)
B) x=–yʹandy=xʹ
C) x=2+2
2xʹ– 2–2
2yʹandy=2–2
2xʹ+2+2
2yʹ
D) x=1
2xʹ–3
2yʹandy=3
2xʹ+1
2yʹ
3) 9x2–4xy+5y2–8x+8y=0
A) x=2–2
2xʹ– 2+2
2yʹandy=2+2
2xʹ+2–2
2yʹ
B) x=2
2(xʹ–yʹ)andy=2
2(xʹ+yʹ)
C) x=–yʹandy=xʹ
D) x=1
2xʹ–3
2yʹandy=3
2xʹ+1
2yʹ
3 AnalyzeanEquationUsingaRotationofAxes
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Rotatetheaxessothatthenewequationcontainsnoxy–term.Discussthenewequation.
1) 24xy–7y2+36=0
A) θ=36.9°
yʹ2
4
–4xʹ2
9
=1
hyperbola
centerat(0,0)
transverseaxisistheyʹ–axis
verticesat(0,±2)
B) θ=53.1°
yʹ2
4
–4xʹ2
9
=1
hyperbola
centerat(0,0)
transverseaxisistheyʹ–axis
verticesat(0,±2)
C) θ=36.9°
yʹ2
9
–xʹ2
16
=1
hyperbola
centerat(0,0)
transverseaxisistheyʹ–axis
verticesat(0,±3)
D) θ=36.9°
4yʹ2
9
–xʹ2
4
=1
hyperbola
centerat(0,0)
transverseaxisistheyʹ–axis
verticesat(0,±3
2)
2) x2+2xy+y2–8x+8y=0
A) θ=45°
xʹ2=–42
yʹ
parabola
vertexat(0,0)
focusat(0,–2)
B) θ=45°
yʹ2=–42xʹ
parabola
vertexat(0,0)
focusat(–2,0)
C) θ=36.9°
xʹ2
4
+yʹ2
4
=1
ellipse
center(0,0)
majoraxisisxʹ–axis
verticesat(±2,0)
D) θ=36.9°
xʹ2
4
+yʹ2
2
=1
ellipse
center(0,0)
majoraxisisxʹ–axis
verticesat(±2,0)
3) 31x2+10 3xy+21y2–144=0
A) θ=30°
xʹ2
4
+yʹ2
9
=1
ellipse
centerat(0,0)
majoraxisisyʹ–axis
verticesat(0,±3)
B) θ=45°
yʹ2=–42
xʹ
parabola
vertexat(0,0)
focusat(–2,0)
C) θ=45°
xʹ2=–42
yʹ
parabola
vertexat(0,0)
focusat(0,–2)
D) θ=36.9°
xʹ2
9
+yʹ2
4
=1
ellipse
centerat(0,0)
majoraxisisxʹ–axis
verticesat(±3,0)
4) xy+16=0
A) θ=45°
yʹ2
32
–xʹ2
32
=1
hyperbola
centerat(0,0)
transverseaxisisyʹ–axis
verticesat(0,±42
)
B) θ=45°
yʹ2=–32xʹ
parabola
vertexat(0,0)
focusat(–8,0)
C) θ=45°
yʹ2
32
+xʹ2
32
=1
ellipse
centerat(0,0)
majoraxisisyʹ–axis
verticesat(0,±42
)
D) θ=36.9°
xʹ2
4
+yʹ2
2
=1
ellipse
centerat(0,0)
majoraxisisthexʹ–axis
verticesat(±2,0)
5) x2+xy+y2–3y–6=0
A) θ=45°
xʹ–2
2
2
5
+
yʹ–32
2
2
15
=1
ellipse
centerat(2
2,32
2)
majoraxisisyʹ–axis
verticesat(2
2,–32
2)and(2
2,92
2)
B) θ=45°
yʹ2=–18xʹ
parabola
vertexat(0,0)
focusat(–9
2,0)
C) θ=45°
xʹ2
6
–yʹ2
8
=1
hyperbola
centerat(0,0)
transverseaxisisthexʹ–axis
verticesat(±6,0)
D) θ=45°
xʹ2
3
+yʹ2
4
=1
ellipse
centerat(0,0)
majoraxisisyʹ–axis
verticesat(0,±2)
6) 17x2–12xy+8y2–68x+24y–12=0
A) θ=63.4°
xʹ–25
5
2
16
+
yʹ+45
5
2
4
=1
ellipse
centerat(25
5,–45
5)
majoraxisisxʹ–axis
verticesat(4+25
5,–45
5)and(–4+25
5,–45
5)
B) θ=63.4°
xʹ2=–16yʹ
parabola
vertexat(0,0)
focusat(0,–4)
C) θ=63.4°
xʹ2
16
–yʹ2
4
=1
hyperbola
centerat(0,0)
transverseaxisisthexʹ–axis
verticesat(±4,0)
D) θ=26.6°
xʹ2
4
+yʹ2
16
=1
ellipse
centerat(0,0)
majoraxisisyʹ–axis
verticesat(0,±4)
7) 5x2–6xy+5y2–8=0
A) θ=45°
xʹ2
4
+yʹ2=1
ellipse
centerat(0,0)
majoraxisisthexʹ–axis
verticesat(±2,0)
B) θ=45°
yʹ2=–4xʹ
parabola
vertexat(0,0)
focusat(–1,0)
C) θ=45°
xʹ2=–4yʹ
parabola
vertexat(0,0)
focusat(0,–1)
D) θ=45°
xʹ2
4
–yʹ2=1
hyperbola
centerat(0,0)
transverseaxisisthexʹ–axis
verticesat(±2,0)
Rotatetheaxessothatthenewequationcontainsnoxy–term.Graphthenewequation.
8) 24xy–7y2+36=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page70
9) x2+2xy+y2–8x+8y=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
10) 31x2+10 3xy+21y2–144=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page72
11) xy+16=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
12) x2+xy+y2–3y–6=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
13) 17x2–12xy+8y2–68x+24y–12=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
14) 5x2–6xy+5y2–8=0
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 8
y
10
8
6
4
2
-2
-4
-6
-8
-10
4 IdentifyConicswithoutaRotationofAxes
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifytheequationwithoutapplyingarotationofaxes.
1) x2+4xy+4y2+2x+4y–8=0
A) parabola B) ellipse C) hyperbola D) notaconic
2) 2x2+6xy+9y2–2x–2y–1=0
A) ellipse B) parabola C) hyperbola D) notaconic
3) 4x2–10xy+2y2+2x–4y–3=0
A) hyperbola B) ellipse C) parabola D) notaconic
4) x2+3xy–3y2–4x+2y+2=0
A) hyperbola B) ellipse C) parabola D) notaconic
5) 4x2+3xy+2y2–3x+2y–7=0
A) ellipse B) hyperbola C) parabola D) notaconic
6) 9x2–6xy+3y2–2x–2y–3=0
A) ellipse B) hyperbola C) parabola D) notaconic
11.6 PolarEquationsofConics
1 AnalyzeandGraphPolarEquationsofConics
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Identifytheconicthatthepolarequationrepresents.Also,givethepositionofthedirectrix.
1) r=2
1–2cosθ
A) hyperbola,directrixperpendiculartothepolaraxis1 leftofthepole
B) hyperbola,directrixperpendiculartothepolaraxis1 rightofthepole
C) ellipse,directrixperpendiculartothepolaraxis1 leftofthepole
D) ellipse,directrixperpendiculartothepolaraxis1 rightofthepole
2) r=6
3+3sinθ
A) parabola,directrixparalleltothepolaraxis2 abovethepole
B) parabola,directrixperpendiculartothepolaraxis2 rightofthepole
C) hyperbola,directrixparalleltothepolaraxis2 abovethepole
D) hyperbola,directrixperpendiculartothepolaraxis2 rightofthepole
3) r=7
12–4sinθ
A) ellipse,directrixparalleltothepolaraxis7
4
belowthepole
B) ellipse,directrixperpendiculartothepolaraxis7
4
leftofthepole
C) ellipse,directrixperpendiculartothepolaraxis7
4
rightofthepole
D) ellipse,directrixparalleltothepolaraxis7
4
abovethepole
4) r=6
3–4cosθ
A) hyperbola,directrixisperpendiculartothepolaraxisatadistance3
2
unitstotheleftofthepole
B) ellipse,directrixisperpendiculartothepolaraxisatadistance3
2
unitstotherightofthepole
C) ellipse,directrixisperpendiculartothepolaraxisatadistance3unitstotheleftofthepole
D) hyperbola,directrixisperpendiculartothepolaraxisatadistance3unitstotherightofthepole
Discusstheequationandgraphit.
5) r=4
2–2cosθ
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) directrixperpendiculartopolar
axis2leftofpole
focus(0,0),vertex1,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) directrixperpendiculartopolar
axis2rightofpole
focus(0,0),vertex1,0
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) directrixparalleltopolar
axis2abovepole
focus(0,0),vertex1,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) directrixparalleltopolar
axis2belowpole
focus(0,0),vertex1,3π
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
Page78
6) r=8
4–sinθ
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) directrixparalleltopolar
axis8belowpole
center8
15,π
2
vertices8
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) directrixperpendiculartopolar
axis8rightofpole
center–8
15,0
vertices8
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) directrixperpendiculartopolar
axis8leftofpole
center8
15,0
vertices8
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) directrixparalleltopolar
axis8abovepole
center–8
15,π
2
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
7) r=3
2+4sinθ
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) hyperbola;directrixparallelto
thepolaraxis3
4
unitabovethepole
vertices(1
2,π
2),(–3
2,3π
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) hyperbola,directrixperpendicularto
thepolaraxis3
4
unitrightofthepole
vertices(1
2,0),(–3
2,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) ellipse,directrixperpendicularto
thepolaraxis3
2
unitleftofthepole
vertices(3
2,0),(1
2,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
D) ellipse,directrixparallelto
thepolaraxis3
2
unitbelowthepole
vertices(3
2,π
2),(1
2,3π
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
2 ConvertthePolarEquationofaConictoaRectangularEquation
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Convertthepolarequationtoarectangularequation.
1) r=12
3–3cosθ
A) y2=8x+16 B) y2=–8x+16 C) x2=8y+16 D) x2=–8y+16
2) r=12
3+cosθ
A) 8x2+9y2+24x–144=0 B) 10x2+9y2–24x–144=0
C) 9x2+8y2+24y–144=0D)9x
2+9y2+24x–144=0
3) r=8secθ
2secθ+1
A) 3x2+4y2+16x–64=0B)5x
2+4y2–16x–64=0
C) 4x2+3y2+16y–64=0D)4x
2+4y2+16x–64=0
4) r=4secθ
secθ+2
A) 3x2–y2–16x+16=0B)3x
2–y2+16=0
C) 3y2–x2–16y+16=0D)3y
2–x2+16=0
Findapolarequationfortheconic.Afocusisatthepole.
5) e=1;directrixisparalleltothepolaraxis1 abovethepole
A) r=1
1+sinθ B) r=1
1–sinθ C) r=1
1+cosθ D) r=1
1–cosθ
6) e=1
2;directrixisperpendiculartothepolaraxis1totheleftofthepole
A) r=2
4–2cosθ B) r=2
4+2cosθ C) r=4
4–2cosθ D) r=4
4+2cosθ
7) e=5;directrixisperpendiculartothepolaraxis3 totherightofthepole
A) r=15
1+5cosθ B) r=15
1–5cosθ C) r=15
1+5sinθ D) r=15
1–5sinθ
11.7 PlaneCurvesandParametricEquations
1 GraphParametricEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphthecurvewhoseparametricequationsaregiven.
1) x=2t,y=t+1;–2≤t≤3
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
2) x=2t–1,y=t2+3;–4≤t≤4
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
50
-50
x
-10 10
y
50
-50
D)
x
-10 10
y
50
-50
x
-10 10
y
50
-50
3) x=t3+1,y=t3–1;–2≤t≤2
x
-30 30
y
40
-40
x
-30 30
y
40
-40
A)
x
-30 30
y
40
-40
x
-30 30
y
40
-40
B)
x
-30 30
y
40
-40
x
-30 30
y
40
-40
C)
x
-30 30
y
40
-40
x
-30 30
y
40
-40
D)
x
-30 30
y
40
-40
x
-30 30
y
40
-40
Page84
4) x=4sint,y=4cost;0≤t≤2π
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
5) x=4tant,y=3sect;0≤t≤2π
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
6) x=t2,y=t+5;0≤t≤4
x
-20 20
y
10
-10
x
-20 20
y
10
-10
A)
x
-20 20
y
10
-10
x
-20 20
y
10
-10
B)
x
-20 20
y
10
-10
x
-20 20
y
10
-10
C)
x
-20 20
y
10
-10
x
-20 20
y
10
-10
D)
x
-20 20
y
10
-10
x
-20 20
y
10
-10
7) x=–sect,y=tant;–π
2
<t< π
2
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) B)
C) D)
Findtheparametricequationsthatdefinethecurveshown.
8)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
(3, 4)
(7, 2)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
(3, 4)
(7, 2)
A) x=2t+3
,
y=–t+4;0≤t≤2B)x= –2t +3
,
y= –t+4;0≤t≤1
C) x=–2t+3
,
y=t+4;0≤t≤2D)x=2t +3
,
y= –t+4;0≤t≤1
9)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
(-3, 0) (3, 0)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
(-3, 0) (3, 0)
A) x=3sin(π
2
(t–1)),y=–4cos(π
2
(t–1));0≤t≤2
B) x=3sin(π
2
(t–1)),y=–4cos(π
2
(t–1));0≤t≤1
C) x=4sin(π
2
(t–1)),y=–3cos(π
2
(t–1));0≤t≤2
D) x=–4sin(π
2
(t–1)),y=3cos(π
2
(t–1));0≤t≤1
Graphthecurvewhoseparametricequationsaregiven.
10) x=3cost,y=–3sint; π
2
≤t≤3π
2
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
Page89
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 –1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 –1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 –1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 –1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
11) x=t,y=4t+2;0≤t≤4
x
-5 5
y
20
-20
x
-5 5
y
20
-20
A)
x
-5 5
y
20
-20
x
-5 5
y
20
-20
B)
x
-5 5
y
20
-20
x
-5 5
y
20
-20
C)
x
-5 5
y
20
-20
x
-5 5
y
20
-20
D)
x
-5 5
y
20
-20
x
-5 5
y
20
-20
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Theparametricequationsoffourcurvesaregiven.Grapheachofthem,indicatingtheorientation.
12) 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
Page91
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useagraphingutilitytographthecurvedefinedbythegivenparametricequations.
13) x=t+2,y=3t–1;0≤t≤3
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) B)
C) D)
14) x=2t2,y=t+2;–∞<t<∞
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) B)
C) D)
Page93
15) x=3cost,y=2sint;0≤t≤2π
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) B)
C) D)
2 FindaRectangularEquationforaCurveDefinedParametrically
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findarectangularequationfortheplanecurvedefinedbytheparametricequations.
1) x=2t,y=t+3;–2≤t≤3
A) y=1
2x+3;forxin–4≤x≤6B)y= –2x+3;forxin–∞
<
x
<
∞
C) y=1
2x–3;forxin–∞<x<∞D) y=x2+1;forxin–2≤x≤2
2) x=2t–1,y=t2+6;–4≤t≤4
A) y=1
4x2+1
2x+25
4;forxin–9≤x≤7B)y=1
2x2+1;forxin–6≤x≤4
C) y=–1
2x+30;forxin–6≤x≤4D)y=x2+1;forxin–2≤x≤2
3) x=t3+1,y=t3–5;–2≤t≤2
A) y=x–6;forxin–7≤x≤9B)y= – x–6;forxin–7≤x≤9
C) y=–x2;forxin–4≤x≤4D)y=x3;forxin–3≤x≤1
4) x=2sint,y=2cost;0≤t≤2π
A) x2+y2=4;forxin–2≤x≤2B)y
2–x2=4;forxin–∞<x<∞
C) y=a2–x2=4;forxin–∞<x<∞D) y=x2–9;forxin–2≤x≤2
5) x=4tant,y=5sect;0≤t≤2π
A) y2
25
–x2
16
=1;forxin–∞<x<∞B) y2
25
+x2
16
=1;forxin–∞<x<∞
C) y=51+x2
16 ;forxin–∞<x<∞D) y=x2–9;forxin–3≤x≤3
6) x=5cost,y=–2sint;0≤t≤2π
A) 4x2+25y2=100;–5≤x≤5B)4x
2+25y2=1;–1
5
≤x≤1
5
C) 4x2–25y2=100;x≥5D)4x
2–25y2=1;x≥1
2
Solvetheproblem.
7) Thepositionofaprojectilefiredwithaninitialvelocityv0feetpersecondandatanangleθtothe
horizontalattheendoftsecondsisgivenbytheparametricequationsx=(v0cosθ)t,
y=(v0sinθ)t–16t2.Supposetheinitialvelocityis4feetpersecond.Obtaintherectangularequationof
thetrajectoryandidentifythecurve.
A) y=–1x2
cos2θ
+(tanθ)x;parabola B) y=–1x2
cos2θ
+(tanθ)x;ellipse
C) y=1x2
cos2θ
+(cotθ)x;parabola D) y=–1x2
cos2θ
+(cotθ)x;hyperbola
3 UseTimeasaParameterinParametricEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Ronthrowsaballstraightupwithaninitialspeedof50 feetpersecondfromaheightof6feet.Find
parametricequationsthatdescribethemotionoftheballasafunctionoftime.Howlongistheballinthe
air?Whenistheballatitsmaximumheight?Whatisthemaximumheightoftheball?
A) x=0,y=–16t2+50t+6
3.241sec,1.563sec,
45.063feet
B) x=0,y=–16t2+50t+6
6.481sec,1.563sec,
39.063feet
C) x=0,y=–16t2+50t+6
3sec,1.563sec,
6feet
D) x=0,y=–16t2+50t+6
6sec,1.563sec,
294feet
2) Abaseballpitcherthrowsabaseballwithaninitialspeedof129 feetpersecondatanangleof20° tothe
horizontal.Theballleavesthepitcherʹshandataheightof4feet.Findparametricequationsthatdescribe
themotionoftheballasafunctionoftime.Howlongistheballintheair?Whenistheballatits
maximumheight?Whatisthemaximumheightoftheball?
A) x=121.22t,y=–16t2+44.12t+4
2.845sec,1.379sec,
34.415feet
B) x=121.22t,y=–16t2+44.12t+4
5.691sec,1.379sec,
30.415feet
C) x=121.22t,y=–16t2+44.12t+4
2.664sec,1.379sec,
3.985feet
D) x=121.22t,y=–16t2+44.12t+4
5.327sec,1.379sec,
247.322feet
3) Abaseballplayerhitabaseballwithaninitialspeedof170 feetpersecondatanangleof40° tothe
horizontal.Theballwashitataheightof4feetofftheground.Findparametricequationsthatdescribethe
motionoftheballasafunctionoftime.Howlongistheballintheair?Whenistheballatitsmaximum
height?Whatisthedistancetheballtraveled?
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
4) Rachelʹsbusleavesat5:25PMandacceleratesattherateof4 meterspersecondpersecond.Rachel,who
canrun7meterspersecond,arrivesatthebusstation3secondsafterthebushasleft.Findparametric
equationsthatdescribethemotionsofthebusandRachelasafunctionoftime.Determinealgebraically
whetherRachelwillcatchthebus.Ifso,when?
A) Bus:x1=2t2,y1=2;Rachel:x2=7(t–3),y2=4
Rachelwonʹtcatchthebus.
B) Bus:x1=2t2,y1=2;Rachel:x2=7(t+3),y2=4
Rachelwonʹtcatchthebus.
C) Bus:x1=2t2,y1=2;Rachel:x2=7(t–3),y2=4
Rachelwillcatchthebusat5:30PM
D) Bus:x1=4t2,y1=2;Rachel:x2=7
2(t–3),y2=4
Rachelwillcatchthebusat5:29PM
5) Rachelʹsbusleavesat1:15PMandacceleratesattherateof4 meterspersecondpersecond.Rachel,who
canrun4meterspersecond,arrivesatthebusstation6secondsafterthebushasleft.Findparametric
equationsthatdescribethemotionsofthebusandRachelasafunctionoftime,andsimulatethemotionof
thebusandRachelbysimultaneouslygraphingtheseequations.
x
100
y
5
4
3
2
1
x
100
y
5
4
3
2
1
A) Bus:x1=2t2,y1=2;
Rachel:x2=4(t–6),y2=4
x
100
y
5
4
3
2
1
x
100
y
5
4
3
2
1
B) Bus:x1=2t2,y1=2;
Rachel:x2=4(t+6),y2=4
x
100
y
5
4
3
2
1
x
100
y
5
4
3
2
1
C) Bus:x1=4t2,y1=2;
Rachel:x2=4(t–6),y2=4
x
100
y
5
4
3
2
1
x
100
y
5
4
3
2
1
D) Bus:x1=4t2,y1=2;
Rachel:x2=4(t–6),y2=4
x
100
y
5
4
3
2
1
x
100
y
5
4
3
2
1
Page97
6) CarA(travellingnorthat30mph)andcarB(travelingwestat70 mph)areheadingtowardthesame
intersection.CarAis4milesfromtheintersectionwhencarBis5milesfromtheintersection.Find
parametricequationsthatdescribethemotionofcarsAandB.
5mi carB70mph
4mi
carA30mph
A) CarA:x=0,y=30t–4;CarB:x=5–70t,y=0
B) CarA:x=–30t+4
,
y=0;CarB:x=5–70t,y=0
C) CarA:x=0,y=70t–5;CarB:x=30t–4
,
y=0
D) CarA:x=70t–5
,
y=0;CarB:x=0,y=4–30t
7) CarA(travellingnorthat30mph)andcarB(travelingwestat20 mph)areheadingtowardthesame
intersection.CarAis3milesfromtheintersectionwhencarBis4milesfromtheintersection..Finda
formulaforthedistancebetweenthecarsasafunctionoftime,usingtheparametricequationsthat
describethemotionofcarsAandB.Usingagraphingutility,findtheminimumdistancebetweenthe
cars.Whenarethecarsclosest?
4mi carB20mph
3mi
carA30mph
A) d=(30t–3)2+(4–20t)2;1.66mi;7.85min
B) d=(30t–3)2+(4–20t)2;1.66mi;0.13min
C) d=(30t–3)2+(4–20t)2;278.19mi;7.85min
D) d=(30t+3)2+(4–20t)2;1.66mi;7.85h
4 FindParametricEquationsforCurvesDefinedbyRectangularEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findparametricequationsfortherectangularequation.
1) y=3x–5
A) x=t,y=3t–5;0≤t
<
∞B) y=3t,3x=t+5;0≤t
<
∞
C) x=t
3,y=t–5
3;0≤t<∞D) x=t,y=3t2–5;0≤t<∞
2) y=x4–2
A) x=t,y=t4–2;0≤t<∞B) x=t2,y=t2–2;0≤t<∞
C) x=t2,y=t4–2;0≤t<∞D) x=t,y=t2–2;0≤t<∞
3) y=3x2+6
A) x=t;y=3t2+6;0≤t<∞B) y=t;x=3t2+6;0≤t<∞
C) x=t2;y=3t+6;0≤t<∞D) x=t;y=3t+6;t≥0
Findtwosetsofparametricequationsforthegivenrectangularequation.
4) y=7x+9
A) x=t,y=7t+9;x=t
7,y=t+9B)x=t,y=7t+9;x=7t,y=t+9
C) x=t,y=7t+9;x=t,y=t
7
+9D)x=7t,y=t+9;x=t
7,y=t+9
Solvetheproblem.
5) Findparametricequationsforanobjectthatmovesalongtheellipsex2
9
+y2
4
=1withthemotion
described.
Themotionbeginsat(0,2),iscounterclockwise,andrequires3secondsforacompleterevolution.
A) x=–3sin(2
3
πt),y=2cos(2
3
πt),0≤t≤3B)x=–3cos(2
3
πt),y=2sin(2
3
πt),0≤t≤3
C) x=3sin(2
3
πt),y=–2cos(2
3
πt),0≤t≤3D)x= –3 sin(3πt),y=2cos(3πt),0≤t≤3
Ch.11 AnalyticGeometry
AnswerKey
11.1 Conics
1 KnowtheNamesoftheConics