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10.8.1: Review Exercises

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    116449
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    Review Exercises

    The Ellipse

    For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

    1.

    x 2 25 + y 2 64 =1 x 2 25 + y 2 64 =1

    2.

    (x2) 2 100 + ( y+3 ) 2 36 =1 (x2) 2 100 + ( y+3 ) 2 36 =1

    3.

    9 x 2 + y 2 +54x4y+76=0 9 x 2 + y 2 +54x4y+76=0

    4.

    9 x 2 +36 y 2 36x+72y+36=0 9 x 2 +36 y 2 36x+72y+36=0

    For the following exercises, graph the ellipse, noting center, vertices, and foci.

    5.

    x 2 36 + y 2 9 =1 x 2 36 + y 2 9 =1

    6.

    (x4) 2 25 + ( y+3 ) 2 49 =1 (x4) 2 25 + ( y+3 ) 2 49 =1

    7.

    4 x 2 + y 2 +16x+4y44=0 4 x 2 + y 2 +16x+4y44=0

    8.

    2 x 2 +3 y 2 20x+12y+38=0 2 x 2 +3 y 2 20x+12y+38=0

    For the following exercises, use the given information to find the equation for the ellipse.

    9.

    Center at ( 0,0 ), ( 0,0 ), focus at ( 3,0 ), ( 3,0 ), vertex at ( −5,0 ) ( −5,0 )

    10.

    Center at ( 2,−2 ), ( 2,−2 ), vertex at ( 7,−2 ), ( 7,−2 ), focus at ( 4,−2 ) ( 4,−2 )

    11.

    A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

    The Hyperbola

    For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.

    12.

    x 2 81 y 2 9 =1 x 2 81 y 2 9 =1

    13.

    ( y+1 ) 2 16 ( x4 ) 2 36 =1 ( y+1 ) 2 16 ( x4 ) 2 36 =1

    14.

    9 y 2 4 x 2 +54y16x+29=0 9 y 2 4 x 2 +54y16x+29=0

    15.

    3 x 2 y 2 12x6y9=0 3 x 2 y 2 12x6y9=0

    For the following exercises, graph the hyperbola, labeling vertices and foci.

    16.

    x 2 9 y 2 16 =1 x 2 9 y 2 16 =1

    17.

    ( y1 ) 2 49 ( x+1 ) 2 4 =1 ( y1 ) 2 49 ( x+1 ) 2 4 =1

    18.

    x 2 4 y 2 +6x+32y91=0 x 2 4 y 2 +6x+32y91=0

    19.

    2 y 2 x 2 12y6=0 2 y 2 x 2 12y6=0

    For the following exercises, find the equation of the hyperbola.

    20.

    Center at ( 0,0 ), ( 0,0 ), vertex at ( 0,4 ), ( 0,4 ), focus at ( 0,−6 ) ( 0,−6 )

    21.

    Foci at ( 3,7 ) ( 3,7 ) and ( 7,7 ), ( 7,7 ), vertex at ( 6,7 ) ( 6,7 )

    The Parabola

    For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

    22.

    y 2 =12x y 2 =12x

    23.

    ( x+2 ) 2 = 1 2 ( y1 ) ( x+2 ) 2 = 1 2 ( y1 )

    24.

    y 2 6y6x3=0 y 2 6y6x3=0

    25.

    x 2 +10xy+23=0 x 2 +10xy+23=0

    For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

    26.

    x 2 +4y=0 x 2 +4y=0

    27.

    ( y1 ) 2 = 1 2 ( x+3 ) ( y1 ) 2 = 1 2 ( x+3 )

    28.

    x 2 8x10y+46=0 x 2 8x10y+46=0

    29.

    2 y 2 +12y+6x+15=0 2 y 2 +12y+6x+15=0

    For the following exercises, write the equation of the parabola using the given information.

    30.

    Focus at ( −4,0 ); ( −4,0 ); directrix is x=4 x=4

    31.

    Focus at ( 2, 9 8 ); ( 2, 9 8 ); directrix is y= 7 8 y= 7 8

    32.

    A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

    Rotation of Axes

    For the following exercises, determine which of the conic sections is represented.

    33.

    16 x 2 +24xy+9 y 2 +24x60y60=0 16 x 2 +24xy+9 y 2 +24x60y60=0

    34.

    4 x 2 +14xy+5 y 2 +18x6y+30=0 4 x 2 +14xy+5 y 2 +18x6y+30=0

    35.

    4 x 2 +xy+2 y 2 +8x26y+9=0 4 x 2 +xy+2 y 2 +8x26y+9=0

    For the following exercises, determine the angle θ θ that will eliminate the xy xy term, and write the corresponding equation without the xy xy term.

    36.

    x 2 +4xy2 y 2 6=0 x 2 +4xy2 y 2 6=0

    37.

    x 2 xy+ y 2 6=0 x 2 xy+ y 2 6=0

    For the following exercises, graph the equation relative to the x y x y system in which the equation has no x y x y term.

    38.

    9 x 2 24xy+16 y 2 80x60y+100=0 9 x 2 24xy+16 y 2 80x60y+100=0

    39.

    x 2 xy+ y 2 2=0 x 2 xy+ y 2 2=0

    40.

    6 x 2 +24xy y 2 12x+26y+11=0 6 x 2 +24xy y 2 12x+26y+11=0

    Conic Sections in Polar Coordinates

    For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

    41.

    r= 10 15cosθ r= 10 15cosθ

    42.

    r= 6 3+2cosθ r= 6 3+2cosθ

    43.

    r= 1 4+3sinθ r= 1 4+3sinθ

    44.

    r= 3 55sinθ r= 3 55sinθ

    For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.

    45.

    r= 3 1sinθ r= 3 1sinθ

    46.

    r= 8 4+3sinθ r= 8 4+3sinθ

    47.

    r= 10 4+5cosθ r= 10 4+5cosθ

    48.

    r= 9 36cosθ r= 9 36cosθ

    For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

    49.

    Directrix is x=3 x=3 and eccentricity e=1 e=1

    50.

    Directrix is y=−2 y=−2 and eccentricity e=4 e=4


    10.8.1: Review Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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