Ch.7 AnalyticGeometry
7.1 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
7.2 TheParabola
1 AnalyzeParabolaswithVertexattheOrigin
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Matchtheequationtoitsgraph.
1) 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
Page2
2) y2=–8x
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=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
Page3
4) x2=–4y
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(–14
,
0);directrixthelinex=14
A) y2=–56x B) y2=56x C) y2=–14x D) x2=–56y
6) Focusat(0,–21);directrixtheliney=21
A) x2=–84y B) x2=84y C) y2=–21x D) y2=–84x
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
,
5)
A) y2=25
8xB)y
2=25
32xC)x
2=25
8yD)x
2=25
32y
Findanequationoftheparaboladescribedandstatethetwopointsthatdefinethelatusrectum.
11) Focusat(0,1);directrixtheliney=–1
A) x2=4y;latusrectum:(2,1)and(–2,1) B) y2=8x;latusrectum:(3,4)and(–3,4)
C) x2=4y;latusrectum:(1,2)and(–1,2) D) x2=8y;latusrectum:(4,1)and(–4,1)
Page4
Findthevertex,focus,anddirectrixoftheparabola.Graphtheequation.
12) x2=–8y
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,–2)
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:(0,0)
focus:(0,2)
directrix:y=–2
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:(–2,0)
directrix:x=2
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:(2,0)
directrix:x=–2
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
Page6
Graphtheequation.
14) y2=5x
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
Page7
15) y2=–18x
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=16y
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
Page9
17) x2=–12y
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=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
Page11
2) (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
Page12
3) (x+2)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
Page13
4) (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
Findanequationfortheparaboladescribed.
5) Vertexat(1
,
2);focusat(4
,
2)
A) (y–2)2=12(x–1) B) (y–2)2=–12(x–1)
C) (x–2)2=8(y–2) D) (x–2)2=–8(y–2)
6) Vertexat(1
,
9);focusat(1
,
7)
A) (x–1)2=–8(y–9) B) (x–1)2=8(y–9)
C) (y–9)2=–24(x–1) D) (y–9)2=24(x–1)
7) Vertexat(5
,
–8);focusat(5
,
–6)
A) (x–5)2=8(y+8) B) (x–5)2=–8(y+8)
C) (y–8)2=4(x+5) D) (y–8)2=–4(x+5)
8) Vertexat(4
,
–3);focusat(7
,
–3)
A) (y+3)2=12(x–4) B) (y+3)2=–12(x–4)
C) (x+4)2=–16(y–3) D) (x+4)2=16(y–3)
Findthevertex,focus,anddirectrixoftheparabolawiththegivenequation.
9) (y+3)2=8(x+1)
A) vertex:(–1
,
–3)
focus:(1,–3)
directrix:x=–3
B) vertex:(1
,
3)
focus:(3,3)
directrix:x=–1
C) vertex:(–3
,
–1)
focus:(–1,–1)
directrix:x=–5
D) vertex:(–1
,
–3)
focus:(–3,–3)
directrix:x=1
Page14
10) (y+4)2=–12(x–3)
A) vertex:(3
,
–4)
focus:(0,–4)
directrix:x=6
B) vertex:(–3
,
4)
focus:(–6,4)
directrix:x=0
C) vertex:(–4
,
3)
focus:(–7,3)
directrix:x=–1
D) vertex:(3
,
–4)
focus:(6,–4)
directrix:x=0
11) (x–2)2=20(y+3)
A) vertex:(2
,
–3)
focus:(2,2)
directrix:y=–8
B) vertex:(–2
,
3)
focus:(–2,8)
directrix:y=–2
C) vertex:(–3
,
2)
focus:(–3,7)
directrix:y=–3
D) vertex:(2
,
–3)
focus:(2,–8)
directrix:x=2
12) (x–3)2=–16(y–2)
A) vertex:(3
,
2)
focus:(3,–2)
directrix:y=6
B) vertex:(–3
,
–2)
focus:(–3,–6)
directrix:y=2
C) vertex:(2
,
3)
focus:(2,–1)
directrix:y=7
D) vertex:(3
,
2)
focus:(3,6)
directrix:x=–2
Findthevertex,focus,anddirectrixoftheparabola.Graphtheequation.
13) (y–3)2=4(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
,
3)
focus:(–2,3)
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:(–3
,
3)
focus:(–2,3)
directrix:x=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page15
C) vertex:(–3
,
3)
focus:(–3,4)
directrix:y=2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(3
,
–3)
focus:(3,–2)
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+3)2=(y+1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(–3
,
–1)
focus:(–3,–0.75)
directrix:y=–1.25
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(3
,
1)
focus:(3,1.25)
directrix:y=0.75
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page16
C) vertex:(–3
,
–1)
focus:(–2.75,–1)
directrix:x=–3.25
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(3
,
1)
focus:(3.25,1)
directrix:x=2.75
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
15) x2–14x=8y–97
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(7
,
6)
focus:(7,8)
directrix:y=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(7
,
6)
focus:(7,4)
directrix:y=8
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page17
C) vertex:(7
,
6)
focus:(9,6)
directrix:x=5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(7
,
6)
focus:(5,6)
directrix:x=9
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
16) y2+10y=4x+3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) vertex:(–7
,
–5)
focus:(–6,–5)
directrix:x=–8
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) vertex:(–7
,
–5)
focus:(–8,–5)
directrix:x=–6
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page18
C) vertex:(–7
,
–5)
focus:(–7,–4)
directrix:y=–6
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) vertex:(–7
,
–5)
focus:(–7,–6)
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+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
Page19
18) (y–2)2=–5(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
Page20
19) (x+1)2=5(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
Page21
20) (x+1)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.Ifthemirroris18 inches
acrossatitsopeningandis2feetdeep,wherewillthelightbeconcentrated?
A) 0.8in.fromthevertex B) 10.1 in.fromthevertex
C) 0.2in.fromthevertex D) 0.1 in.fromthevertex
2) Asearchlightisshapedlikeaparaboloidofrevolution.Ifthelightsourceislocated5feetfromthebase
alongtheaxisofsymmetryandtheopeningis14feetacross,howdeepshouldthesearchlightbe?
A) 2.5ft B) 0.9ft C) 9.8 ft D) 12.3 ft
Page22
3) Abridgeisbuiltintheshapeofaparabolicarch.Thebridgearchhasaspanof166feetandamaximum
heightof25feet.Findtheheightofthearchat15feetfromitscenter.
A) 24.2ft B) 0.2ft C) 3.3 ft D) 29.9 ft
4) Areflectingtelescopehasamirrorshapedlikeaparaboloidofrevolution.Ifthedistanceofthevertexto
thefocusis28feetandthedistanceacrossthetopofthemirroris68inches,howdeepisthemirrorinthe
center?
A) 289
336
in. B) 289
28
in. C) 289
4032
in. D) 98
17
in.
5) Anexperimentalmodelforasuspensionbridgeisbuiltintheshapeofaparabolicarch.Inonesection,
cablerunsfromthetopofonetowerdowntotheroadway,justtouchingitthere,andupagaintothetop
ofasecondtower.Thetowersareboth9inchestallandstand60inchesapart.Findtheverticaldistance
fromtheroadwaytothecableatapointontheroad15inchesfromthelowestpointofthecable.
A) 2.25in. B) 9in. C) 2.45 in. D) 2.05 in.
6) Anexperimentalmodelforasuspensionbridgeisbuiltintheshapeofaparabolicarch.Inonesection,
cablerunsfromthetopofonetowerdowntotheroadway,justtouchingitthere,andupagaintothetop
ofasecondtower.Thetowersareboth6.25inchestallandstand50inchesapart.Atsomepointalongthe
roadfromthelowestpointofthecable,thecableis0.56inchesabovetheroadway.Findthedistance
betweenthatpointandthebaseofthenearesttower.
A) 17.5in. B) 7.3in. C) 17.7 in. D) 7.7in.
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
7.3 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
49
+x2
9
=1B)
y2
49
–x2
9
=1C)
x2
9
–y2
49
=1D)
x2
49
+y2
9
=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
81
+y2
4
=1
A) centerat(0,0)
fociat(–77,0)and(77,0)
verticesat(–9,0),(9,0)
B) centerat(0,0)
fociat(0,–77)and(0,77)
verticesat(0,–9),(0,9)
C) centerat(0,0)
fociat(–9,0)and(9,0)
verticesat(–81,0),(81,0)
D) centerat(0,0)
fociat(0,–2)and(0,2)
verticesat(0,–4),(0,4)
Page24
4) x2
4
+y2
36
=1
A) centerat(0,0)
fociat(0,–42
)and(0,42)
verticesat(0,–6),(0,6)
B) centerat(0,0)
fociat(–42,0)and(4 2,0)
verticesat(–6,0),(6,0)
C) centerat(0,0)
fociat(0,–6)and(0,6)
verticesat(0,–36),(0,36)
D) centerat(0,0)
fociat(0,6)and(2,0)
verticesat(0,36),(4,0)
5) 4x2+81y2=324
A) centerat(0,0)
fociat(–77,0)and(77,0)
verticesat(–9,0),(9,0)
B) centerat(0,0)
fociat(0,–77)and(0,77)
verticesat(0,–9),(0,9)
C) centerat(0,0)
fociat(–9,0)and(9,0)
verticesat(–81,0),(81,0)
D) centerat(0,0)
fociat(0,–2)and(0,2)
verticesat(0,–4),(0,4)
6) 16x2+4y2=64
A) centerat(0,0)
fociat(0,–23)and(0,23)
verticesat(0,–4),(0,4)
B) centerat(0,0)
fociat(–23,0)and(2 3,0)
verticesat(–4,0),(4,0)
C) centerat(0,0)
fociat(0,–4)and(0,4)
verticesat(0,–16),(0,16)
D) centerat(0,0)
fociat(0,4)and(2,0)
verticesat(0,16)and(4,0)
Findanequationfortheellipsedescribed.
7) Centerat(0,0);focusat(6
,
0);vertexat(7
,
0)
A) x2
49
+y2
13
=1B)
x2
13
+y2
49
=1C)
x2
36
+y2
13
=1D)
x2
36
+y2
49
=1
8) 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
9) Centerat(0,0);focusat(–2
,
0);vertexat(3
,
0)
A) x2
9
+y2
5
=1B)
x2
5
+y2
9
=1C)
x2
4
+y2
5
=1D)
x2
4
+y2
9
=1
10) Centerat(0,0);focusat(0,3);vertexat(0,8)
A) x2
55
+y2
64
=1B)
x2
64
+y2
55
=1C)
x2
9
+y2
55
=1D)
x2
9
+y2
64
=1
11) Centerat(0,0);focusat(0,–3);vertexat(0,6)
A) x2
27
+y2
36
=1B)
x2
36
+y2
27
=1C)
x2
9
+y2
27
=1D)
x2
9
+y2
36
=1
12) Centerat(0,0);focusat(0,7);vertexat(0,–8)
A) x2
15
+y2
64
=1B)
x2
64
+y2
15
=1C)
x2
49
+y2
15
=1D)
x2
49
+y2
64
=1
Page25
13) Focusat(–7
,
0);verticesat(±8
,
0)
A) x2
64
+y2
15
=1B)
x2
15
+y2
64
=1C)
x2
49
+y2
15
=1D)
x2
49
+y2
64
=1
14) Focusat(0,–3);verticesat(0,±7)
A) x2
40
+y2
49
=1B)
x2
49
+y2
40
=1C)
x2
9
+y2
40
=1D)
x2
9
+y2
49
=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±4
A) x2
12
+y2
16
=1B)x2
16
+y2
12
=1C)
x2
4
+y2
12
=1D)
x2
4
+y2
16
=1
17) Center(0,0);majoraxishorizontalwithlength10;lengthofminoraxisis4
A) x2
25
+y2
4
=1B)
x2
4
+y2
25
=1C)
x2
10
+y2
4
=1D)
x2
100
+y2
16
=1
18) Center(0,0);majoraxisverticalwithlength12;lengthofminoraxisis6
A) x2
9
+y2
36
=1B)
x2
36
+y2
9
=1C)
x2
6
+y2
36
=1D)
x2
36
+y2
144
=1
Page26
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
Page27
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
Page28
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) 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
Page30
2 AnalyzeEllipseswithCenterat(h,k)
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writeanequationforthegraph.
1)
x
-5 5
y
5
-5
(-2, 1)
x
-5 5
y
5
-5
(-2, 1)
A) (x+2)2
16
+(y–1)2
9
=1B)
(x–1)2
16
+(y+2)2
9
=1
C) (x–2)2
16
+(y+1)2
9
=1D)
(x+2)2
9
+(y–1)2
16
=1
Findthecenter,foci,andverticesoftheellipse.
2) (x+3)2
36
+(y–3)2
16
=1
A) centerat(–3
,
3)
fociat(–3+25
,3),(–3–25,3)
verticesat(–9,3),(3,3)
B) centerat(3
,
–3)
fociat(3+25
,–3),(3–25,–3)
verticesat(–9,3),(3,3)
C) centerat(–3
,
3)
fociat(–25
,3),(2 5,3)
verticesat(6,3),(–6,3)
D) centerat(–3
,
3)
fociat(–3+25
,–3),(–3–25,–3)
verticesat(6,3),(–6,3)
3) 36(x+3)2+16(y+1)2=576
A) centerat(–3
,
–1)
fociat(–3,–1–25
),(–3,–1+25)
verticesat(–3,5),(–3,–7)
B) centerat(–1
,
–3)
fociat(–1,–3–25
),(–1,–3+25)
verticesat(–1,5),(–1,–7)
C) centerat(3
,
–1)
fociat(3,–1–25
),(3,–1+25)
verticesat(3,5),(3,–7)
D) centerat(–2
,
–1)
fociat(–2,–1–25
),(–2,–1+25)
verticesat(–2,5),(–2,–7)
Page31
4) 4x2+5y2–32x+40y+124=0
A) (x–4)2
5
+(y+4)2
4
=1
center:(4,–4);foci:(5,–4),(3,–4);vertices:(6.2,–4),(1.8,–4)
B) (x–4)2
4
+(y+4)2
5
=1
center:(4,–4);foci:(5,–4),(3,–4);vertices:(6.2,–4),(1.8,–4)
C) (x–4)2
5
+(y+4)2
4
=1
center:(–4,4);foci:(–3,4),(–5,4);vertices:(–6.2,4),(–1.8,4)
D) (x–4)2
4
+(y+4)2
5
=1
center:(–4,4);foci:(–3,4),(–5,4);vertices:(–6.2,4),(–1.8,4)
5) 9x2+y2–144x+567=0
A) (x–8)2+y2
9
=1
center:(8,0);foci:(8,22
),(8,–22);vertices:(8,3),(8,–3)
B) x2
9
+(y–8)2=1
center:(8,0);foci:(8,22
),(8,–22);vertices:(8,3),(8,–3)
C) (x–3)2+y2
64
=1
center:(3,0);foci:(3,37
),(3,–37);vertices:(3,8),(3,–8)
D) x2
64
+(y–3)2=1
center:(3,0);foci:(3,37
),(3,–22);vertices:(3,8),
(3,–8)
Page32
Graphtheequation.
6) (x–1)2
9
+(y+1)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
Page33
7) (x+2)2
9
+(y+1)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–1)2+9(y+2)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
Page35
9) 16(x–2)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
Findanequationfortheellipsedescribed.
10) Centerat(2
,
4);focusat(7
,
4);vertexat(9
,
4)
A) (x–2)2
49
+(y–4)2
24
=1B)
(x+2)2
49
+(y+4)2
24
=1
C) (x–2)2
100
+(y+4)2
9
=2D)
(x+2)2
25
–(y–4)2
19
=1
11) Verticesat(–2
,
4)and(8
,
4);focusat(6
,
4)
A) (x–3)2
25
+(y–4)2
16
=1B)
(x–4)2
16
+(y–3)2
15
=1
C) (x+3)2
9
+(y+4)2
16
=1D)
(x–3)2
49
–(y+4)2
19
=1
Page36
Findanequationfortheellipsedescribed.Graphtheequation.
12) Fociat(4
,
1)and(4
,
–5);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+2)2
25
+(x–4)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+2)2
25
+(y+4)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+2)2
25
+(x+4)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+2)2
16
+(y+4)2
25
=1
Page37
13) Fociat(–4
,
–1)and(2
,
–1);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+1)2
25
+(y+1)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+4)2
25
+(x+1)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–4)2
25
+(x–1)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
25
+(y–1)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
Page39
15) Fociat(–2
,
4)and(–8
,
4);vertexat(–9
,
4)
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–4)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–4)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–4)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–4)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
Page40
16) Centerat(–2
,
3);focusat(–8
,
3);containsthepoint(–10
,
3)
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–3)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–3)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+3)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+3)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
Page41
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.Ithasaspanof116 feet.Theheightofthearch30 feet
fromthecenteristobe11feet.Findtheheightofthearchatitscenter.
A) 12.85ft B) 11.39 ft C) 30.55 ft D) 21.27 ft
3) Anarchforabridgeoverahighwayisintheformofasemiellipse.Thetopofthearchis30 feetabove
ground(themajoraxis).Whatshouldthespanofthebridgebe(thelengthofitsminoraxis)iftheheight
26feetfromthecenteristobe14feetaboveground?
A) 58.79ft B) 29.4 ft C) 111.43 ft D) 56.12 ft
Page42
4) Theorbitofaplanetaroundasunisanellipsewiththesunatonefocus.Theaphelionofaplanetisits
greatestdistancefromthesun,itsperihelionisitsshortestdistance,anditsmeandistanceisthelengthof
thesemimajoraxisoftheellipticalorbit.Ifaplanethasaperihelionof514.6millionmilesandamean
distanceof517millionmiles,writeanequationfortheorbitoftheplanetaroundthesun.
A) x2
5172
+y2
516.9942
=1B)
x2
517.0062
+y2
5172
=1
C) x2
5172
+y2
2.42
=1D)
x2
5172
+y2
514.62
=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
7.4 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
Page44
2) y2
4
–x2
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
Findanequationforthehyperboladescribed.
3) Verticesat(0,±4);asymptotesaty=±2
9x
A) y2
16
–x2
324
=1B)
y2
324
–x2
16
=1C)
y2
16
–x2
81
=1D)
y2
81
–x2
4
=1
4) Verticesat(±7
,
0);fociat(±8
,
0)
A) x2
49
–y2
15
=1B)
x2
15
–y2
49
=1C)
x2
49
–y2
64
=1D)
x2
64
–y2
49
=1
Page45
Findanequationforthehyperboladescribed.Graphtheequation.
5) Centerat(0,0);focusat(13
,0);vertexat(2,0)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x2
4
–y2
9
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) 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
C) 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
D) y2
4
–x2
9
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page46
6) Centerat(0,0);vertexat(0,2);focusat(0,29)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y2
4
–x2
25
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) y2
25
–x2
4
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) x2
25
–y2
4
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) x2
4
–y2
25
=1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page47
Findthecenter,transverseaxis,vertices,foci,andasymptotesofthehyperbola.
7) x2
144
–y2
64
=1
A) centerat(0,0)
transverseaxisisx–axis
verticesat(–12,0)and(12,0)
fociat(–413
,0)and(4 13,0)
asymptotesofy=–2
3
andy=2
3
B) centerat(0,0)
transverseaxisisx–axis
verticesat(–8,0)and(8,0)
fociat(–413
,0)and(4 13,0)
asymptotesofy=–2
3
andy=2
3
C) centerat(0,0)
transverseaxisisy–axis
verticesat(0,–12)and(0,12)
fociat(–413,0)and(4 13,0)
asymptotesofy=–2
3
andy=2
3
D) centerat(0,0)
transverseaxisisx–axis
verticesat(–12,0)and(12,0)
fociat(–8,0)and(8,0)
asymptotesofy=–2
3
andy=2
3
8) 121y2–100x2=12,100
A) centerat(0,0)
transverseaxisisy–axis
verticesat(0,–10)and(0,10)
fociat(0,–221)and(0,221)
asymptotesofy=–10
11
andy=10
11
B) centerat(0,0)
transverseaxisisx–axis
vertices:(–11,0),(11,0)
foci:(–221,0),( 221,0)
asymptotesofy=–10
11
andy=10
11
C) centerat(0,0)
transverseaxisisy–axis
vertices:(0,–10),(0,10)
foci:(–221,0),( 221,0)
asymptotesofy=–10
11
andy=10
11
D) centerat(0,0)
transverseaxisisx–axis
vertices:(–10,0),(10,0)
foci:(–11,0),(11,0)
asymptotesofy=–10
11
andy=10
11
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
9
–y2
25
=1B)
y2
9
–x2
25
=1C)
x2
25
–y2
9
=1D)
y2
25
–x2
9
=1
Page48
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
9
–x2
4
=1B)
x2
9
–y2
4
=1C)
x2
4
–y2
9
=1D)
y2
4
–x2
9
=1
Page49
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
4
–x2
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
Page51
13) 16x2–9y2=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
Page52
14) 16y2–9x2=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
Page53
15) 16x2=9y2+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
Page54
16) 16y2=4x2+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
2 FindtheAsymptotesofaHyperbola
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheasymptotesofthehyperbola.
1) x2
16
–y2
4
=1
A) y=1
2xandy=–1
2xB)y=2xandy= – 2x
C) y=1
4xandy=–1
4xD)y=4xandy= – 4x
Page55
2) y2–x2=4
A) y=xandy=–xB)y=2xandy= – 2x
C) y=1
2xandy=–1
2xD)y=1
4xandy=–1
4x
3) (x+3)2
9
–(y+1)2
4
=1
A) y+1=2
3(x+3)andy+1=–2
3(x+3) B) y+1=3
2(x+3)andy+1=–3
2(x+3)
C) y=2
3(x+3)andy=–2
3(x+3) D) y+3=2
3(x+1)andy+3=–2
3(x+1)
4) x2–y2–4x–2y–13=0
A) y+1=(x–2)andy+1=–(x–2) B) y+1=1
4(x–2)andy+1=–1
4(x–2)
C) y+2=(x–1)andy+2=–(x–1) D) y–2=(x+1)andy–2=–(x+1)
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
,
6);focusat(1
,
6);vertexat(3
,
6)
A) (x–4)2–(y–6)2
8
=1B)
(x–4)2
8
–(y–6)2=1
C) (x–6)2–(y–4)2
8
=1D)
(x–6)2
8
–(y–4)2=1
3) Verticesat(0,±8);asymptotesaty=±4
9x
A) y2
64
–x2
324
=1B)
y2
324
–x2
64
=1C)y2
64
–x2
81
=1D)
y2
81
–x2
16
=1
4) Verticesat(±8
,
0);fociat(±11
,
0)
A) x2
64
–y2
57
=1B)
x2
57
–y2
64
=1C)
x2
64
–y2
121
=1D)
x2
121
–y2
64
=1
Page56
Findthecenter,transverseaxis,vertices,foci,andasymptotesofthehyperbola.
5) (x–1)2
4
–(y–4)2
16
=1
A) centerat(1
,
4)
transverseaxisisparalleltox–axis
verticesat(–1,4)and(3,4)
fociat(1–25
,4)and(1+25,4)
asymptotesofy–4=–2(x–1)andy–4=2(x–1)
B) centerat(4
,
1)
transverseaxisisparalleltox–axis
verticesat(2,1)and(6,1)
fociat(4–25,1)and(4+25,1)
asymptotesofy–1=–2(x–4)andy–1=2(x–4)
C) centerat(1
,
4)
transverseaxisisparalleltoy–axis
verticesat(1,2)and(1,6)
fociat(1,4–25)and(1,4+25)
asymptotesofy+4=–1
2(x+1)andy+4=1
2(x+1)
D) centerat(1
,
4)
transverseaxisisparalleltox–axis
verticesat(–3,4)and(5,4)
fociat(1–25,4)and(1+25,4)
asymptotesofy–4=–1
2(x–1)andy–4=1
2(x–1)
6) (x+1)2–4(y–2)2=4
A) centerat(–1
,
2)
transverseaxisisparalleltox–axis
verticesat(–3,2)and(1,2)
fociat(–1–5,2)and(–1+5,2)
asymptotesofy–2=–1
2(x+1)andy–2=1
2(x+1)
B) centerat(2
,
–1)
transverseaxisisparalleltox–axis
verticesat(0,–1)and(4,–1)
fociat(2–5,–1)and(2+5,–1)
asymptotesofy+1=–1
2(x–2)andy+1=1
2(x–2)
C) centerat(–1
,
2)
transverseaxisisparalleltoy–axis
verticesat(–1,0)and(–1,4),
fociat(–1,2–5)and(–1,2+5),
asymptotesofy+2=–2(x–1)andy+2=2(x–1)
D) centerat(–1
,
2)
transverseaxisisparalleltox–axis
verticesat(–2,2)and(0,2)
fociat(–1–5,2)and(–1+5,2)
asymptotesofy–2=–2(x+1)andy–2=2(x+1)
Page57
7) x2–16y2–8x+96y–144=0
A) centerat(4
,
3)
transverseaxisisparalleltox–axis
verticesat(0,3)and(8,3)
fociat(4–17,3)and(4+17,3)
asymptotesofy–3=–1
4(x–4)andy–3=1
4(x–4)
B) centerat(3
,
4)
transverseaxisisparalleltox–axis
verticesat(–1,4)and(7,4)
fociat(3–17,4)and(3+17,4)
asymptotesofy–4=–1
4(x–3)andy–4=1
4(x–3)
C) centerat(4
,
3)
transverseaxisisparalleltoy–axis
verticesat(4,–1)and(4,7)
fociat(4,3–17)and(4,3+17)
asymptotesofy+3=–4(x+4)andy+3=4(x+4)
D) centerat(4
,
3)
transverseaxisisparalleltox–axis
verticesat(3,3)and(5,3)
fociat(4–17,3)and(4+17,3)
asymptotesofy–3=–4(x–4)andy–3=4(x–4)
Page58
Graphthehyperbola.
8) (x+2)2
4
–(y–2)2
25
=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
Page59
9) (y+2)2
9
–(x–2)2
25
=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
Page60
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
Page61
11) (y+2)2–4(x–3)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) Tworecordingdevicesareset3000feetapart,withthedeviceatpointAtothewestofthedeviceatpoin
t
B.Atapointonalinebetweenthedevices,200feetfrompointB,asmallamountofexplosiveis
detonated.Therecordingdevicesrecordthetimethesoundreacheseachone.Howfardirectlynorthof
siteBshouldasecondexplosionbedonesothatthemeasuredtimedifferencerecordedbythedevicesis
thesameasthatforthefirstdetonation?
A) 430.77ft B) 5562.37 ft C) 1486.61 ft D) 1142.61 ft
Page62
2) Theroofofabuildingisintheshapeofthehyperbolay2–x2=12,wherexandyareinmeters.Referto
thefigureanddeterminetheheighthoftheoutsidewalls.
a=b=3m
A) 4.6mB)21mC)
–3mD)9m
3) Theroofofabuildingisintheshapeofthehyperbolay2–x2=61,wherexandyareinmeters.
Determinethedistance,w,theoutsidewallsareapart,iftheheightofeachwallis11m.
A) 15.5mB)60m C) 7.75 m D) 13.5 m
4) Acometfollowsthehyperbolicpathdescribedbyx2
19
–y2
3
=1,wherexandyareinmillions.Ifthesunis
thefocusofthepath,howclosetothesunisthevertexofthepath?
A) 0.3million B) 4.7million C) 4.4 million D) 22million
Page63
5) Asatellitefollowingthehyperbolicpathshowninthepictureturnsrapidlyat(0,6)andthenmovescloser
andclosertotheliney=4xasitgetsfartherfromthetrackingstationattheorigin.Findtheequationthat
describesthepathoftherocketifthecenterofthehyperbolaisat(0,0).
(0,6)
y=4x
A) y2
36
–x2
9
4
=1B)
x2
36
–y2
24
3
2
=1C)
y2
9
4
–x2
36
=1D)
x2
24
3
2
–y2
36
=1
Page64
Ch.7 AnalyticGeometry
AnswerKey
7.1 Conics
1 KnowtheNamesoftheConics
Page65
Page67