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8.6 Review Exercises and Sample Exam

Review Exercises

    Distance, Midpoint, and the Parabola

      Calculate the distance and midpoint between the given two points.

    1. (0,2) and (4,1)

    2. (6,0) and (2,6)

    3. (2,4) and (6,8)

    4. (12,1) and (52,12)

    5. (0,32) and (5,42)

    6. (53,6) and (33,6)

      Determine the area of a circle whose diameter is defined by the given two points.

    1. (3,3) and (3,3)

    2. (2,9) and (10,15)

    3. (23,12) and (13,32)

    4. (25,22) and (0,42)

      Rewrite in standard form and give the vertex.

    1. y=x210x+33

    2. y=2x24x1

    3. y=x23x1

    4. y=x2x2

    5. x=y2+10y+10

    6. x=3y2+12y+7

    7. x=y2+8y3

    8. x=5y25y+2

      Rewrite in standard form and graph. Be sure to find the vertex and all intercepts.

    1. y=x220x+75

    2. y=x210x+75

    3. y=2x212x24

    4. y=4x2+4x+6

    5. x=y210y+16

    6. x=y2+4y+12

    7. x=4y2+12y

    8. x=9y2+18y+12

    9. x=4y2+4y+2

    10. x=y25y+2

    Circles

      Determine the center and radius given the equation of a circle in standard form.

    1. (x6)2+y2=9

    2. (x+8)2+(y10)2=1

    3. x2+y2=5

    4. (x38)2+(y+52)2=12

      Determine standard form for the equation of the circle:

    1. Center (7,2) with radius r=10.

    2. Center (13,1) with radius r=23.

    3. Center (0,5) with radius r=27.

    4. Center (1,0) with radius r=532.

    5. Circle whose diameter is defined by (4,10) and (2,8).

    6. Circle whose diameter is defined by (3,6) and (0,4).

      Find the x- and y-intercepts.

    1. (x3)2+(y+5)2=16

    2. (x+5)2+(y1)2=4

    3. x2+(y2)2=20

    4. (x3)2+(y+3)2=8

    5. x2+y212y+27=0

    6. x2+y24x+2y+1=0

      Graph.

    1. (x+8)2+(y6)2=4

    2. (x20)2+(y+152)2=2254

    3. x2+y2=24

    4. (x1)2+y2=14

    5. x2+(y7)2=27

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

      Rewrite in standard form and graph.

    1. x2+y26x+4y3=0

    2. x2+y2+8x10y+16=0

    3. 2x2+2y22x6y3=0

    4. 4x2+4y2+8y+1=0

    5. x2+y25x+y12=0

    6. x2+y2+12x8y=0

    Ellipses

      Given the equation of an ellipse in standard form, determine its center, orientation, major radius, and minor radius.

    1. (x+12)216+(y10)24=1

    2. (x+3)23+y225=1

    3. x2+(y5)212=1

    4. (x8)25+(y+8)18=1

      Determine the standard form for the equation of the ellipse given the following information.

    1. Center (0,4) with a=3 and b=4.

    2. Center (3,8) with a=1 and b=7.

    3. Center (0,0) with a=5 and b=2.

    4. Center (10,30) with a=10 and b=1.

      Find the x- and y-intercepts.

    1. (x+2)24+y29=1

    2. (x1)22+(y+1)23=1

    3. 5x2+2y2=20

    4. 5(x3)2+6y2=120

      Graph.

    1. (x10)225+(y+5)24=1

    2. (x+6)29+(y8)236=1

    3. (x32)24+(y72)2=1

    4. (x23)2+y24=1

    5. x22+y25=1

    6. (x+2)28+(y3)212=1

      Rewrite in standard form and graph.

    1. 4x2+9y28x+90y+193=0

    2. 9x2+4y2+108x80y+580=0

    3. x2+9y2+6x+108y+324=0

    4. 25x2+y2350x8y+1,216=0

    5. 8x2+12y216x36y13=0

    6. 10x2+2y250x+14y+7=0

    Hyperbolas

      Given the equation of a hyperbola in standard form, determine its center, which way the graph opens, and the vertices.

    1. (x10)24(y+5)216=1

    2. (x+7)22(y8)28=1

    3. (y20)23(x15)2=1

    4. 3y212(x1)2=36

      Determine the standard form for the equation of the hyperbola.

    1. Center (25,10), a=3, b=5, opens up and down.

    2. Center (9,12), a=53, b=7, opens left and right.

    3. Center (4,0), a=1, b=6, opens left and right.

    4. Center (2,3), a=102, b=23, opens up and down.

      Find the x- and y-intercepts.

    1. (x1)24(y+3)29=1

    2. (x+4)28(y2)212=1

    3. 4(y2)2x2=16

    4. 6(y+1)23(x1)2=18

      Graph.

    1. (x10)225(y+5)2100=1

    2. (x4)24(y8)216=1

    3. (y3)29(x6)281=1

    4. (y+1)24(x+1)225=1

    5. y227(x3)29=1

    6. x22y23=1

      Rewrite in standard form and graph.

    1. 4x29y28x90y257=0

    2. 9x2y2108x+16y+224=0

    3. 25y22x2100y+50=0

    4. 3y2x22x10=0

    5. 8y212x2+24y12x33=0

    6. 4y24x216y28x37=0

      Identify the conic sections and rewrite in standard form.

    1. x2+y22x8y+16=0

    2. x2+2y2+4x24y+74=0

    3. x2y26x4y+3=0

    4. x2+y10x+22=0

    5. x2+12y212x+24=0

    6. x2+y2+10y+22=0

    7. 4y220x2+16y+20x9=0

    8. 16x16y2+24y25=0

    9. 9x29y26x18y17=0

    10. 4x2+4y2+4x8y+1=0

      Given the graph, write the equation in general form.

    Solving Nonlinear Systems

      Solve.

    1. {x2+y2=8xy=4
    2. {x2+y2=1x+2y=1
    3. {x2+3y2=42xy=1
    4. {2x2+y2=5x+y=3
    5. {3x22y2=1xy=2
    6. {x23y2=10x2y=1
    7. {2x2+y2=114x+y2=5
    8. {x2+4y2=12x2+4y=5
    9. {5x2y2=10x2+y=2
    10. {2x2+y2=12x4y2=3
    11. {x2+4y2=10xy=2
    12. {y+x2=0xy8=0
    13. {1x+1y=101x1y=6
    14. {1x+1y=1yx=2
    15. {x2y2=3y=x4
    16. {(x1)2+y2=1yx=0

Answers

  1. Distance: 5 units; midpoint: (2,12)

  2. Distance: 410 units; midpoint: (4,2)

  3. Distance: 7 units; midpoint: (52,722)

  4. 18π square units

  5. 5π4 square units

  6. y=(x5)2+8; vertex: (5,8)

  7. y=(x32)2134; vertex: (32,134)

  8. x=(y+5)215; vertex: (15,5)

  9. x=(y4)2+13; vertex: (13,4)

  10. y=(x10)225;

  11. y=2(x+3)26;

  12. x=(y5)29;

  13. x=4(y32)2+9;

  14. x=4(y12)2+3;

  1. Center: (6,0); radius: r=3

  2. Center: (0,0); radius: r=5

  3. (x+7)2+(y2)2=100

  4. x2+(y+5)2=28

  5. (x+3)2+(y9)2=2

  6. x-intercepts: none; y-intercepts: (0,5±7)

  7. x-intercepts: (±4,0); y-intercepts: (0,2±25)

  8. x-intercepts: none; y-intercepts: (0,3), (0,9)

  9. (x3)2+(y+2)2=16;

  10. (x12)2+(y32)2=4;

  11. (x52)2+(y+12)2=7;

  1. Center: (12,10); orientation: horizontal; major radius: 4 units; minor radius: 2 units

  2. Center: (0,5); orientation: vertical; major radius: 23 units; minor radius: 1 unit

  3. x29+(y+4)216=1

  4. x225+y22=1

  5. x-intercepts: (4,0), (0,0); y-intercepts: (0,0)

  6. x-intercepts: (±2,0); y-intercepts: (0,±10)

  7. (x1)29+(y+5)24=1;

  8. (x+3)29+(y+6)2=1;

  9. (x1)26+(y32)24=1;

  1. Center: (10,5); opens left and right; vertices: (8,5), (12,5)

  2. Center: (15,20); opens upward and downward; vertices: (15,203), (15,20+3)

  3. (y10)25(x+25)29=1

  4. (x+4)2y236=1

  5. x-intercepts: (1±22,0); y-intercepts: none

  6. x-intercepts: (0,0); y-intercepts: (0,0), (0,4)

  7. (x1)29(y+5)24=1;

  8. (y2)22x225=1;

  9. (y+32)26(x+12)24=1;

  10. Circle; (x1)2+(y4)2=1

  11. Hyperbola; (x3)22(y+2)22=1

  12. Ellipse; (x6)212+y2=1

  13. Hyperbola; (y+2)25(x12)2=1

  14. Hyperbola; (x13)2(y+1)2=1

  15. x2+y2+18x6y+9=0

  16. 9x2y2+72x12y+72=0

  17. 9x2+64y2+54x495=0

  1. (2,2)

  2. (113,1513), (1,1)

  3. (9,11), (1,1)

  4. (1,3), (1,3)

  5. (2,0), (2,0), (7,5), (7,5)

  6. (2,2), (2,2), (22,22), (22,22)

  7. (18,12)

  8. (5,1)

Sample Exam

    1. Given two points (4,6) and (2,8):

      1. Calculate the distance between them.
      2. Find the midpoint between them.
    2. Determine the area of a circle whose diameter is defined by the points (4,3) and (1,2).

      Rewrite in standard form and graph. Find the vertex and all intercepts if any.

    1. y=x2+6x5

    2. x=2y2+4y6

    3. x=3y2+3y+1

    4. Find the equation of a circle in standard form with center (6,3) and radius 25 units.

      Sketch the graph of the conic section given its equation in standard form.

    1. (x4)2+(y+1)2=45

    2. (x+3)24+y29=1

    3. y23x29=1

    4. x216(y2)2=1

      Rewrite in standard form and graph.

    1. 9x2+4y2144x+16y+556=0

    2. xy2+6y+7=0

    3. x2+y2+20x20y+100=0

    4. 4y2x2+40y30x225=0

      Find the x- and y-intercepts.

    1. x=2(y4)2+9

    2. (y1)212(x+1)2=1

      Solve.

    1. {x+y=2y=x2+4
    2. {yx2=3x2+y2=9
    3. {2xy=1(x+1)2+2y2=1
    4. {x2+y2=6xy=3
    5. Find the equation of an ellipse in standard form with vertices (3,5) and (5,5) and a minor radius 2 units in length.

    6. Find the equation of a hyperbola in standard form opening left and right with vertices (±5,0) and a conjugate axis that measures 10 units.

    7. Given the graph of the ellipse, determine its equation in general form.

    8. A rectangular deck has an area of 80 square feet and a perimeter that measures 36 feet. Find the dimensions of the deck.

    9. The diagonal of a rectangle measures 213 centimeters and the perimeter measures 20 centimeters. Find the dimensions of the rectangle.

Answers

    1. 210 units;
    2. (1,7)
  1. y=(x3)2+4;

  2. x=3(y12)2+74;

  3. (x8)24+(y+2)29=1;

  4. (x+10)2+(y10)2=100;

  5. x-intercept: (23,0); y-intercepts: (0,8±322)

  6. (1,3), (2,0)

  7. Ø

  8. (x1)216+(y+5)24=1

  9. 4x2+25y224x100y+36=0

  10. 6 centimeters by 4 centimeters