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13.2.10: Chapter 10

  • Page ID
    117285
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    Try It

    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)

    b2ec109544091d745a7ae836c7cf35db54011ee34.

    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)

    0e737b089bdd8054841a8294d67ccfdeeff6f2eb5.

    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)

    fd84b64f7b9f310503d634df0bc5b7146a0a49a76.

    (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.

    1. ⓐ x257,600+y225,600=1x257,600+y225,600=1
    2. ⓑ 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;

    c315d2b489a8535d6b6bd7810ded932996bf11f35.

    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

    f55d78608dc0c90c104e6a49e2caa8a203d3ec036.

    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)

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    Focus: (0,2);(0,2); Directrix: y=−2;y=−2; Endpoints of the latus rectum: (±4,2).(±4,2).

    fb2d5a2609085d48d5efff756bfcbb7212b7c80d3.

    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).

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    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).

    7297ffaacad2091ab1448f047075519325daafef6.

    1. ⓐ y2=1280xy2=1280x
    2. ⓑ The depth of the cooker is 500 mm

    10.4 Rotation of Axes

    1.

    1. ⓐ hyperbola
    2. ⓑ ellipse

    2.

    x′24+y′21=1x′24+y′21=1

    3.

    1. ⓐ hyperbola
    2. ⓑ ellipse

    10.5 Conic Sections in Polar Coordinates

    1.

    ellipse; e=13;x=−2e=13;x=−2

    2.

    4a575f682c34a9d263f0c395a4e40f14f9a4b994

    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)

    a82343b66fbae11e1cdefa2af7807ae4b021935e35.

    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)

    462bcd587f176701310b6ef873fb2b68bab7f3c837.

    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.

    ec26f05aeb9d624c260734b203b8f6e41499215b39.

    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)

    1556bf0a07d37a8ad3e504de3c28f86cada6e0ce41.

    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)

    30e9d0de308eff10347674e76d3046fc433b0b1743.

    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)

    f3d602734c8ddff2d3e0bc7dfac7c1161c3bbcde45.

    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)

    ca4ab27be15a293c54cddf3034f6372247046bb447.

    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.

    fdf6c3f2cbab7f8dee97789fe86fdf24d4e1694b

    33.

    54d1247006162f2e403bd70c80648a6ff3b49043

    35.

    03a3f623d0d44a6acff3d2c286117c3257e23d25

    37.

    3244ba9a0895805241eede69173a4c04dcab90cd

    39.

    5461b9d196e49a2da673af59061f233c21bc8e5b

    41.

    4c8b4cf4a99c74175532412a7bdd37f56569400a

    43.

    ebb78cf02271a24caf2fd955b8c8a4f3ff7841a9

    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

    cd14d7f489d4ccc82593bd0107eed53f2351f83b59.

    y(x)=1+2x2+4x+5−−−−−−−−−√,y(x)=1−2x2+4x+5−−−−−−−−−√y(x)=1+2x2+4x+5,y(x)=1−2x2+4x+5

    0c89c8c4010bed5dbff59647221ba92b78cd3b1761.

    x225−y225=1x225−y225=1

    b1369bb79cc265b54f7540771cf6e366e317eb7163.

    x2100−y225=1x2100−y225=1

    94d71c6d4a5025a1941f3073aa36296e5dc24cc765.

    x2400−y2225=1x2400−y2225=1

    c2944b00fb78c091c1004129d33273d63c350df967.

    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.

    87f03c7644e4801d4b687b6d66ae591f570026d6

    33.

    73ced13bac4e2882aef12168e7259c91a7af7603

    35.

    ff1769bd2efa0c894e105160760af7578d257fa0

    37.

    da9c60f65c54dd12c45bc111afa137f6019beb12

    39.

    3d83d25d75e50810486c080f27bbb4bcbf70af2a

    41.

    d17cdffa57fd836e867c009ee9132f9dde896351

    43.

    0710b58cba97ea10acd70b6154e2ddbdda693687

    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

    6dfbb83afe961ca0596078c9956dbc79321bdfbf33.

    (x′−y′)28+(x′+y′)22=1(x′−y′)28+(x′+y′)22=1

    f9567d3f8610ada96b529721a0e3a7f885c6c5f735.

    (x′+y′)22−(x′−y′)22=1(x′+y′)22−(x′−y′)22=1

    16a8dc7a0c91fa720bf38b60258a525228ca1ce437.

    3√2x′−12y′=(12x′+3√2y′−1)232x′−12y′=(12x′+32y′−1)2

    0f8a127ab0d1d92ee1ba9f74f5e7d685f22f3e1239.

    d61d20f268f293feb330cd594f7093dfe9e07ee6

    41.

    254eac23511aacad4f5ed93fc5b6c4c997685cd3

    43.

    63435ab794e839984f36abed6535fdc65e838646

    45.

    0aa59907af779f09cf2c7f2f1b6e28ea12eb2bad

    47.

    3d28abebf73c594f3716f13f57d5c0f03494d7f3

    49.

    b801c153d19fdebb8a380f90fdc1537a80abf9ab

    51.

    θ=45∘θ=45∘

    498d436b6b6c1ca917e669fa252c797fa1a9677053.

    θ=60∘θ=60∘

    023617cfa53a2a36fa9828ebf63880766626c9aa55.

    θ≈36.9∘θ≈36.9∘

    6a3f4621b2ae4c93354f6e47b238cdc7789999fa57.

    −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.

    038c28cff1c24a0d41cab9a4bbc3b1cbebcfc596

    33.

    3189655a5a94b104c7153374b6096c080255ac30

    35.

    6e9344b2f87ac1ea8294801c3900188d01accde7

    37.

    5600759557d1c7da78884945773f3607fdad67ec

    39.

    126ddd055120076e8a9b7b02c2a35717b5dbe591

    41.

    f300b66b92797349e4ef1268ad4723d3e6565bd7

    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)

    c080be43df7edfa438788778d0dfb36e9807b15a7.

    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)

    8c4bc9ce464677072df07dd21e342c394f5f3b729.

    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.

    14011370bb7e706d3a44406f51d4a7df7817ae2119.

    29b012d8db268d6179759aae32838eaee384d22f21.

    (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.

    b71e1c477d32d4d74fdee8e88b9933f6cf1123e729.

    c35e207ae4b9969f15caf1ad8221ba209440a0bb31.

    (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∘

    fe5c56cda2095a5098b339ee3638507a1ef8011e41.

    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.

    a20b3fc13b553bf1692a5ca36dee29c99b11ba7e47.

    f462c254f631e4fedb723bf1af6fc8acad5b50fd49.

    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)

    d67b05139eac21c725bbc640004e31558f5463bf5.

    (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

    eaa9361c5af97ff641ce1120321df4470291c9ac11.

    (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.

    4ddde24b5f86b3509827cd8575d9f899d2f39f4d17.

    Approximately 8.498.49 feet

    19.

    parabola; θ≈63.4∘θ≈63.4∘

    21.

    x′2−4x′+3y′=0x′2−4x′+3y′=0

    4b516d594ce21e55f34d07e49f3ac3b5ce87335b23.

    Hyperbola with e=32,e=32, and directrix 5656 units to the right of the pole.

    25.

    cac72f4524e481daa679ab9d108219f1de829d65


    13.2.10: Chapter 10 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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