13.2.10: Chapter 10
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10.1 The Ellipse
1.
x2+y216=1x2+y216=1
2.
(x−1)216+(y−3)24=1(x−1)216+(y−3)24=1
3.
center: (0,0);(0,0); vertices: (±6,0);(±6,0); co-vertices: (0,±2);(0,±2); foci: (±42–√,0)(±42,0)
4.
Standard form: x216+y249=1;x216+y249=1; center: (0,0);(0,0); vertices: (0,±7);(0,±7); co-vertices: (±4,0);(±4,0); foci: (0,±33−−√)(0,±33)
5.
Center: (4,2);(4,2); vertices: (−2,2)(−2,2) and (10,2);(10,2); co-vertices: (4,2−25–√)(4,2−25) and (4,2+25–√);(4,2+25); foci: (0,2)(0,2) and (8,2)(8,2)
6.
(x−3)24+(y+1)216=1;(x−3)24+(y+1)216=1; center: (3,−1);(3,−1); vertices: (3,−5)(3,−5) and (3,3);(3,3); co-vertices: (1,−1)(1,−1) and (5,−1);(5,−1); foci: (3,−1−23–√)(3,−1−23) and (3,−1+23–√)(3,−1+23)
7.
- ⓐ x257,600+y225,600=1x257,600+y225,600=1
- ⓑ The people are standing 358 feet apart.
10.2 The Hyperbola
1.
Vertices: (±3,0);(±3,0); Foci: (±34−−√,0)(±34,0)
2.
y24−x216=1y24−x216=1
3.
(y−3)225+(x−1)2144=1(y−3)225+(x−1)2144=1
4.
vertices: (±12,0);(±12,0); co-vertices: (0,±9);(0,±9); foci: (±15,0);(±15,0); asymptotes: y=±34x;y=±34x;
5.
center: (3,−4);(3,−4); vertices: (3,−14)(3,−14) and (3,6);(3,6); co-vertices: (−5,−4);(−5,−4); and (11,−4);(11,−4); foci: (3,−4−241−−√)(3,−4−241) and (3,−4+241−−√);(3,−4+241); asymptotes: y=±54(x−3)−4y=±54(x−3)−4
6.
The sides of the tower can be modeled by the hyperbolic equation. x2400−y23600=1or x2202−y2602=1.x2400−y23600=1or x2202−y2602=1.
10.3 The Parabola
1.
Focus: (−4,0);(−4,0); Directrix: x=4;x=4; Endpoints of the latus rectum: (−4,±8)(−4,±8)
2.
Focus: (0,2);(0,2); Directrix: y=−2;y=−2; Endpoints of the latus rectum: (±4,2).(±4,2).
3.
x2=14y.x2=14y.
4.
Vertex: (8,−1);(8,−1); Axis of symmetry: y=−1;y=−1; Focus: (9,−1);(9,−1); Directrix: x=7;x=7; Endpoints of the latus rectum: (9,−3)(9,−3) and (9,1).(9,1).
5.
Vertex: (−2,3);(−2,3); Axis of symmetry: x=−2;x=−2; Focus: (−2,−2);(−2,−2); Directrix: y=8;y=8; Endpoints of the latus rectum: (−12,−2)(−12,−2) and (8,−2).(8,−2).
6.
- ⓐ y2=1280xy2=1280x
- ⓑ The depth of the cooker is 500 mm
10.5 Conic Sections in Polar Coordinates
1.
ellipse; e=13;x=−2e=13;x=−2
2.
3.
r=11−cosθr=11−cosθ
4.
4−8x+3x2−y2=04−8x+3x2−y2=0
10.1 Section Exercises
1.
An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.
3.
This special case would be a circle.
5.
It is symmetric about the x-axis, y-axis, and the origin.
7.
yes; x232+y222=1x232+y222=1
9.
yes; x2(12)2+y2(13)2=1x2(12)2+y2(13)2=1
11.
x222+y272=1;x222+y272=1; Endpoints of major axis (0,7)(0,7) and (0,−7).(0,−7). Endpoints of minor axis (2,0)(2,0) and (−2,0).(−2,0). Foci at (0,35–√),(0,−35–√).(0,35),(0,−35).
13.
x2(1)2+y2(13)2=1;x2(1)2+y2(13)2=1; Endpoints of major axis (1,0)(1,0) and (−1,0).(−1,0). Endpoints of minor axis (0,13),(0,−13).(0,13),(0,−13). Foci at (22√3,0),(−22√3,0).(223,0),(−223,0).
15.
(x−2)272+(y−4)252=1;(x−2)272+(y−4)252=1; Endpoints of major axis (9,4),(−5,4).(9,4),(−5,4). Endpoints of minor axis (2,9),(2,−1).(2,9),(2,−1). Foci at (2+26–√,4),(2−26–√,4).(2+26,4),(2−26,4).
17.
(x+5)222+(y−7)232=1;(x+5)222+(y−7)232=1; Endpoints of major axis (−5,10),(−5,4).(−5,10),(−5,4). Endpoints of minor axis (−3,7),(−7,7).(−3,7),(−7,7). Foci at (−5,7+5–√),(−5,7−5–√).(−5,7+5),(−5,7−5).
19.
(x−1)232+(y−4)222=1;(x−1)232+(y−4)222=1; Endpoints of major axis (4,4),(−2,4).(4,4),(−2,4). Endpoints of minor axis (1,6),(1,2).(1,6),(1,2). Foci at (1+5–√,4),(1−5–√,4).(1+5,4),(1−5,4).
21.
(x−3)2(32√)2+(y−5)2(2√)2=1;(x−3)2(32)2+(y−5)2(2)2=1; Endpoints of major axis (3+32–√,5),(3−32–√,5). (3+32,5),(3−32,5). Endpoints of minor axis (3,5+2–√),(3,5−2–√). (3,5+2),(3,5−2). Foci at (7,5),(−1,5).(7,5),(−1,5).
23.
(x+5)2(5)2+(y−2)2(2)2=1;(x+5)2(5)2+(y−2)2(2)2=1; Endpoints of major axis (0,2),(−10,2).(0,2),(−10,2). Endpoints of minor axis (−5,4),(−5,0).(−5,4),(−5,0). Foci at (−5+21−−√,2),(−5−21−−√,2).(−5+21,2),(−5−21,2).
25.
(x+3)2(5)2+(y+4)2(2)2=1;(x+3)2(5)2+(y+4)2(2)2=1; Endpoints of major axis (2,−4),(−8,−4).(2,−4),(−8,−4). Endpoints of minor axis (−3,−2),(−3,−6).(−3,−2),(−3,−6). Foci at (−3+21−−√,−4),(−3−21−−√,−4).(−3+21,−4),(−3−21,−4).
27.
Foci (−3,−1+11−−√),(−3,−1−11−−√)(−3,−1+11),(−3,−1−11)
29.
Focus (0,0)(0,0)
31.
Foci (−10,30),(−10,−30)(−10,30),(−10,−30)
33.
Center (0,0),(0,0), Vertices (4,0),(−4,0),(0,3),(0,−3),(4,0),(−4,0),(0,3),(0,−3), Foci (7–√,0),(−7–√,0)(7,0),(−7,0)
35.
Center (0,0),(0,0), Vertices (19,0),(−19,0),(0,17),(0,−17), (19,0),(−19,0),(0,17),(0,−17), Foci (0,42√63),(0,−42√63)(0,4263),(0,−4263)
37.
Center (−3,3),(−3,3), Vertices (0,3),(−6,3),(−3,0),(−3,6),(0,3),(−6,3),(−3,0),(−3,6), Focus (−3,3)(−3,3)
Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.
39.
Center (1,1),(1,1), Vertices (5,1),(−3,1),(1,3),(1,−1),(5,1),(−3,1),(1,3),(1,−1), Foci (1,1+23–√),(1,1−23–√)(1,1+23),(1,1−23)
41.
Center (−4,5),(−4,5), Vertices (−2,5),(−6,4),(−4,6),(−4,4),(−2,5),(−6,4),(−4,6),(−4,4), Foci (−4+3–√,5),(−4−3–√,5)(−4+3,5),(−4−3,5)
43.
Center (−2,1),(−2,1), Vertices (0,1),(−4,1),(−2,5),(−2,−3),(0,1),(−4,1),(−2,5),(−2,−3), Foci (−2,1+23–√),(−2,1−23–√)(−2,1+23),(−2,1−23)
45.
Center (−2,−2),(−2,−2), Vertices (0,−2),(−4,−2),(−2,0),(−2,−4),(0,−2),(−4,−2),(−2,0),(−2,−4), Focus (−2,−2)(−2,−2)
47.
x225+y229=1x225+y229=1
49.
(x−4)225+(y−2)21=1(x−4)225+(y−2)21=1
51.
(x+3)216+(y−4)24=1(x+3)216+(y−4)24=1
53.
x281+y29=1x281+y29=1
55.
(x+2)24+(y−2)29=1(x+2)24+(y−2)29=1
57.
Area = 12πsquareunitsArea = 12πsquareunits
59.
Area = 25–√πArea = 25π square units.
61.
Area = 9πArea = 9π square units.
63.
x24h2+y214h2=1x24h2+y214h2=1
65.
x2400+y2144=1x2400+y2144=1 . Distance = 17.32 feet
67.
Approximately 51.96 feet
10.2 Section Exercises
1.
A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.
3.
The foci must lie on the transverse axis and be in the interior of the hyperbola.
5.
The center must be the midpoint of the line segment joining the foci.
7.
yes x262−y232=1x262−y232=1
9.
yes x242−y252=1x242−y252=1
11.
x252−y262=1;x252−y262=1; vertices: (5,0),(−5,0);(5,0),(−5,0); foci: (61−−√,0),(−61−−√,0);(61,0),(−61,0); asymptotes: y=65x,y=−65xy=65x,y=−65x
13.
y222−x292=1;y222−x292=1; vertices: (0,2),(0,−2);(0,2),(0,−2); foci: (0,85−−√),(0,−85−−√);(0,85),(0,−85); asymptotes: y=29x,y=−29xy=29x,y=−29x
15.
(x−1)232−(y−2)242=1;(x−1)232−(y−2)242=1; vertices: (4,2),(−2,2);(4,2),(−2,2); foci: (6,2),(−4,2);(6,2),(−4,2); asymptotes: y=43(x−1)+2,y=−43(x−1)+2y=43(x−1)+2,y=−43(x−1)+2
17.
(x−2)272−(y+7)272=1;(x−2)272−(y+7)272=1; vertices: (9,−7),(−5,−7);(9,−7),(−5,−7); foci: (2+72–√,−7),(2−72–√,−7);(2+72,−7),(2−72,−7); asymptotes: y=x−9,y=−x−5y=x−9,y=−x−5
19.
(x+3)232−(y−3)232=1;(x+3)232−(y−3)232=1; vertices: (0,3),(−6,3);(0,3),(−6,3); foci: (−3+32–√,1),(−3−32–√,1);(−3+32,1),(−3−32,1); asymptotes: y=x+6,y=−xy=x+6,y=−x
21.
(y−4)222−(x−3)242=1;(y−4)222−(x−3)242=1; vertices: (3,6),(3,2);(3,6),(3,2); foci: (3,4+25–√),(3,4−25–√);(3,4+25),(3,4−25); asymptotes: y=12(x−3)+4,y=−12(x−3)+4y=12(x−3)+4,y=−12(x−3)+4
23.
(y+5)272−(x+1)2702=1;(y+5)272−(x+1)2702=1; vertices: (−1,2),(−1,−12);(−1,2),(−1,−12); foci: (−1,−5+7101−−−√),(−1,−5−7101−−−√);(−1,−5+7101),(−1,−5−7101); asymptotes: y=110(x+1)−5,y=−110(x+1)−5y=110(x+1)−5,y=−110(x+1)−5
25.
(x+3)252−(y−4)222=1;(x+3)252−(y−4)222=1; vertices: (2,4),(−8,4);(2,4),(−8,4); foci: (−3+29−−√,4),(−3−29−−√,4);(−3+29,4),(−3−29,4); asymptotes: y=25(x+3)+4,y=−25(x+3)+4y=25(x+3)+4,y=−25(x+3)+4
27.
y=25(x−3)−4,y=−25(x−3)−4y=25(x−3)−4,y=−25(x−3)−4
29.
y=34(x−1)+1,y=−34(x−1)+1y=34(x−1)+1,y=−34(x−1)+1
31.
33.
35.
37.
39.
41.
43.
45.
x29−y216=1x29−y216=1
47.
(x−6)225−(y−1)211=1(x−6)225−(y−1)211=1
49.
(x−4)225−(y−2)21=1(x−4)225−(y−2)21=1
51.
y216−x225=1y216−x225=1
53.
y29−(x+1)29=1y29−(x+1)29=1
55.
(x+3)225−(y+3)225=1(x+3)225−(y+3)225=1
57.
y(x)=3x2+1−−−−−√,y(x)=−3x2+1−−−−−√y(x)=3x2+1,y(x)=−3x2+1
59.
y(x)=1+2x2+4x+5−−−−−−−−−√,y(x)=1−2x2+4x+5−−−−−−−−−√y(x)=1+2x2+4x+5,y(x)=1−2x2+4x+5
61.
x225−y225=1x225−y225=1
63.
x2100−y225=1x2100−y225=1
65.
x2400−y2225=1x2400−y2225=1
67.
4(x−1)2−y22=164(x-1)2-y22=16
69.
(x−h)2a2−(y−k)2b2=(x−3)2−9y2=4(x−h)2a2=4-(y-k)2b2=(x-3)2-9y2=4
10.3 Section Exercises
1.
A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.
3.
The graph will open down.
5.
The distance between the focus and directrix will increase.
7.
yes x2=4(116)yx2=4(116)y
9.
yes (y−3)2=4(2)(x−2)(y−3)2=4(2)(x−2)
11.
y2=18x,V:(0,0);F:(132,0);d:x=−132y2=18x,V:(0,0);F:(132,0);d:x=−132
13.
x2=−14y,V:(0,0);F:(0,−116);d:y=116x2=−14y,V:(0,0);F:(0,−116);d:y=116
15.
y2=136x,V:(0,0);F:(1144,0);d:x=−1144y2=136x,V:(0,0);F:(1144,0);d:x=−1144
17.
(x−1)2=4(y−1),V:(1,1);F:(1,2);d:y=0(x−1)2=4(y−1),V:(1,1);F:(1,2);d:y=0
19.
(y−4)2=2(x+3),V:(−3,4);F:(−52,4);d:x=−72(y−4)2=2(x+3),V:(−3,4);F:(−52,4);d:x=−72
21.
(x+4)2=24(y+1),V:(−4,−1);F:(−4,5);d:y=−7(x+4)2=24(y+1),V:(−4,−1);F:(−4,5);d:y=−7
23.
(y−3)2=−12(x+1),V:(−1,3);F:(−4,3);d:x=2(y−3)2=−12(x+1),V:(−1,3);F:(−4,3);d:x=2
25.
(x−5)2=45(y+3),V:(5,−3);F:(5,−145);d:y=−165(x−5)2=45(y+3),V:(5,−3);F:(5,−145);d:y=−165
27.
(x−2)2=−2(y−5),V:(2,5);F:(2,92);d:y=112(x−2)2=−2(y−5),V:(2,5);F:(2,92);d:y=112
29.
(y−1)2=43(x−5),V:(5,1);F:(163,1);d:x=143(y−1)2=43(x−5),V:(5,1);F:(163,1);d:x=143
31.
33.
35.
37.
39.
41.
43.
45.
x2=−16yx2=−16y
47.
(y−2)2=42–√(x−2)(y−2)2=42(x−2)
49.
(y+3–√)2=−42–√(x−2–√)(y+3)2=−42(x−2)
51.
x2=yx2=y
53.
(y−2)2=14(x+2)(y−2)2=14(x+2)
55.
(y−3–√)2=45–√(x+2–√)(y−3)2=45(x+2)
57.
y2=−8xy2=−8x
59.
(y+1)2=12(x+3)(y+1)2=12(x+3)
61.
(0,1)(0,1)
63.
At the point 2.25 feet above the vertex.
65.
0.5625 feet
67.
x2=−125(y−20),x2=−125(y−20), height is 7.2 feet
69.
2304 feet
10.4 Section Exercises
1.
The xyxy term causes a rotation of the graph to occur.
3.
The conic section is a hyperbola.
5.
It gives the angle of rotation of the axes in order to eliminate the xyxy term.
7.
AB=0,AB=0, parabola
9.
AB=−4<0,AB=−4<0, hyperbola
11.
AB=6>0,AB=6>0, ellipse
13.
B2−4AC=0,B2−4AC=0, parabola
15.
B2−4AC=0,B2−4AC=0, parabola
17.
B2−4AC=−96<0,B2−4AC=−96<0, ellipse
19.
7x′2+9y′2−4=07x′2+9y′2−4=0
21.
3x′2+2x′y′−5y′2+1=03x′2+2x′y′−5y′2+1=0
23.
θ=60∘,11x′2−y′2+3–√x′+y′−4=0θ=60∘,11x′2−y′2+3x′+y′−4=0
25.
θ=150∘,21x′2+9y′2+4x′−43–√y′−6=0θ=150∘,21x′2+9y′2+4x′−43y′−6=0
27.
θ≈36.9∘,125x′2+6x′−42y′+10=0θ≈36.9∘,125x′2+6x′−42y′+10=0
29.
θ=45∘,3x′2−y′2−2–√x′+2–√y′+1=0θ=45∘,3x′2−y′2−2x′+2y′+1=0
31.
2√2(x′+y′)=12(x′−y′)222(x′+y′)=12(x′−y′)2
33.
(x′−y′)28+(x′+y′)22=1(x′−y′)28+(x′+y′)22=1
35.
(x′+y′)22−(x′−y′)22=1(x′+y′)22−(x′−y′)22=1
37.
3√2x′−12y′=(12x′+3√2y′−1)232x′−12y′=(12x′+32y′−1)2
39.
41.
43.
45.
47.
49.
51.
θ=45∘θ=45∘
53.
θ=60∘θ=60∘
55.
θ≈36.9∘θ≈36.9∘
57.
−46–√<k<46–√−46<k<46
59.
k=2k=2
10.5 Section Exercises
1.
If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.
3.
The directrix will be parallel to the polar axis.
5.
One of the foci will be located at the origin.
7.
Parabola with e=1e=1 and directrix 3434 units below the pole.
9.
Hyperbola with e=2e=2 and directrix 5252 units above the pole.
11.
Parabola with e=1e=1 and directrix 310310 units to the right of the pole.
13.
Ellipse with e=27e=27 and directrix 22 units to the right of the pole.
15.
Hyperbola with e=53e=53 and directrix 115115 units above the pole.
17.
Hyperbola with e=87e=87 and directrix 7878 units to the right of the pole.
19.
25x2+16y2−12y−4=025x2+16y2−12y−4=0
21.
21x2−4y2−30x+9=021x2−4y2−30x+9=0
23.
64y2=48x+964y2=48x+9
25.
96y2−25x2+110y+25=096y2−25x2+110y+25=0
27.
3x2+4y2−2x−1=03x2+4y2−2x−1=0
29.
5x2+9y2−24x−36=05x2+9y2−24x−36=0
31.
33.
35.
37.
39.
41.
43.
r=45+cosθr=45+cosθ
45.
r=41+2sinθr=41+2sinθ
47.
r=11+cosθr=11+cosθ
49.
r=78−28cosθr=78−28cosθ
51.
r=122+3sinθr=122+3sinθ
53.
r=154−3cosθr=154−3cosθ
55.
r=33−3cosθr=33−3cosθ
57.
r=±21+sinθcosθ√r=±21+sinθcosθ
59.
r=±24cosθ+3sinθr=±24cosθ+3sinθ
Review Exercises
1.
x252+y282=1;x252+y282=1; center: (0,0);(0,0); vertices: (5,0),(−5,0),(0,8),(0,−8);(5,0),(−5,0),(0,8),(0,−8); foci: (0,39−−√),(0,−39−−√)(0,39),(0,−39)
3.
(x+3)212+(y−2)232=1(−3,2);(−2,2),(−4,2),(−3,5),(−3,−1);(−3,2+22–√),(−3,2−22–√)(x+3)212+(y−2)232=1(−3,2);(−2,2),(−4,2),(−3,5),(−3,−1);(−3,2+22),(−3,2−22)
5.
center: (0,0);(0,0); vertices: (6,0),(−6,0),(0,3),(0,−3);(6,0),(−6,0),(0,3),(0,−3); foci: (33–√,0),(−33–√,0)(33,0),(−33,0)
7.
center: (−2,−2);(−2,−2); vertices: (2,−2),(−6,−2),(−2,6),(−2,−10);(2,−2),(−6,−2),(−2,6),(−2,−10); foci: (−2,−2+43–√,),(−2,−2−43–√)(−2,−2+43,),(−2,−2−43)
9.
x225+y216=1x225+y216=1
11.
Approximately 35.71 feet
13.
(y+1)242−(x−4)262=1;(y+1)242−(x−4)262=1; center: (4,−1);(4,−1); vertices: (4,3),(4,−5);(4,3),(4,−5); foci: (4,−1+213−−√),(4,−1−213−−√)(4,−1+213),(4,−1−213)
15.
(x−2)222−(y+3)2(23√)2=1;(x−2)222−(y+3)2(23)2=1; center: (2,−3);(2,−3); vertices: (4,−3),(0,−3);(4,−3),(0,−3); foci: (6,−3),(−2,−3)(6,−3),(−2,−3)
17.
19.
21.
(x−5)21−(y−7)23=1(x−5)21−(y−7)23=1
23.
(x+2)2=12(y−1);(x+2)2=12(y−1); vertex: (−2,1);(−2,1); focus: (−2,98);(−2,98); directrix: y=78y=78
25.
(x+5)2=(y+2);(x+5)2=(y+2); vertex: (−5,−2);(−5,−2); focus: (−5,−74);(−5,−74); directrix: y=−94y=−94
27.
29.
31.
(x−2)2=(12)(y−1)(x−2)2=(12)(y−1)
33.
B2−4AC=0,B2−4AC=0, parabola
35.
B2−4AC=−31<0,B2−4AC=−31<0, ellipse
37.
θ=45∘,x′2+3y′2−12=0θ=45∘,x′2+3y′2−12=0
39.
θ=45∘θ=45∘
41.
Hyperbola with e=5e=5 and directrix 22 units to the left of the pole.
43.
Ellipse with e=34e=34 and directrix 1313 unit above the pole.
45.
47.
49.
r=31+cos θr=31+cos θ
Practice Test
1.
x232+y222=1;x232+y222=1; center: (0,0);(0,0); vertices: (3,0),(–3,0),(0,2),(0,−2);(3,0),(–3,0),(0,2),(0,−2); foci: (5–√,0),(−5–√,0)(5,0),(−5,0)
3.
center: (3,2);(3,2); vertices: (11,2),(−5,2),(3,8),(3,−4);(11,2),(−5,2),(3,8),(3,−4); foci: (3+27–√,2),(3−27–√,2)(3+27,2),(3−27,2)
5.
(x−1)236+(y−2)227=1(x−1)236+(y−2)227=1
7.
x272−y292=1;x272−y292=1; center: (0,0);(0,0); vertices (7,0),(−7,0);(7,0),(−7,0); foci: (130−−−√,0),(−130−−−√,0);(130,0),(−130,0); asymptotes: y=±97xy=±97x
9.
center: (3,−3);(3,−3); vertices: (8,−3),(−2,−3);(8,−3),(−2,−3); foci: (3+26−−√,−3),(3−26−−√,−3);(3+26,−3),(3−26,−3); asymptotes: y=±15(x−3)−3y=±15(x−3)−3
11.
(y−3)21−(x−1)28=1(y−3)21−(x−1)28=1
13.
(x−2)2=13(y+1);(x−2)2=13(y+1); vertex: (2,−1);(2,−1); focus: (2,−1112);(2,−1112); directrix: y=−1312y=−1312
15.
17.
Approximately 8.498.49 feet
19.
parabola; θ≈63.4∘θ≈63.4∘
21.
x′2−4x′+3y′=0x′2−4x′+3y′=0
23.
Hyperbola with e=32,e=32, and directrix 5656 units to the right of the pole.
25.
