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3.2e: Circle Exercises.

  • Page ID
    55753
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    A: Write the Equation for a Circle from a Description

    Exercise \(\PageIndex{A}\) 

    1. Write an equation of the circle centered at (8 , -10) with radius 8.

    2. Write an equation of the circle centered at (-9, 9) with radius 16.

    Answer

    1. \( (x-8)^2+(y+10)^2 = 64 \)

    B: Construct an equation for a circle from a graph

    Exercise \(\PageIndex{B}\) 

    \( \bigstar \) State the center and radius of the circle  graphed below and construct the equation for it.

    3. 
    circleGraph1.png
    4.
    circleGraph2.png
    5.
    circleGraph3.png
    6.
    circleGraph5.png
    Answers to Odd Numbered Problems:
    3. Center \(  (2, 1) \), radius  \(  r = 2 \), \( (x - 2)^{2} + (y + 1)^{2} = 4 \) 5. Center \(  (-1, 3) \), radius  \(  r = 5 \), \( (x+1)^{2} + (y -3)^{2} = 25 \)  

    C: Graph a circle given an equation

    Exercise \(\PageIndex{C}\) 

    \( \bigstar \) State the center and radius of the circle described by the equation and graph it.

    1. \(\left(x-2\right)^{2} + \left(y+3\right)^{2} = 9\)
    2. \(\left(x+1\right)^{2} + \left(y-2\right)^{2} = 16\)
    3. \( (x - 2)^{2} + (y + 5)^{2} = 4 \) 
    4. \( (x + 1)^{2} + (y + 5)^{2} = 100 \)
    5. \( (x + 9)^{2} + y^{2} = 25 \)
    6. \( (x-4)^2+(y+2)^2 = 9 \) 
    7. \( (x+4)^2 + (y-5)^2 = 42 \) 
    1. \( (x-3)^2+(y-6)^2 = 20 \)  
    2. \( \left(x + \frac{5}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 = \frac{30}{4} \)  
    3. \( (x-1)^2 + (y-5)^2 = 5 \)  
    4. \( \left(x + \frac{1}{2}\right)^{2} + \left(y - \frac{3}{5}\right)^{2} = \frac{161}{100} \)   
    5. \( (x - 3)^{2} + (y - 5)^{2} = 65 \)  
    6. \( x^{2} + (y - 3)^{2} = 0 \)  
    7. \( x^{2} + (y - 72)^{2} = 4096 \)
    Answers to Odd Numbered Problems:
    11. Center \(  (2, -3) \), radius  \(  r = 3 \)
    13. Center \(  (2, -5) \), radius  \(  r = 2 \)
    15. Center \(  (-9, 0) \), radius  \(  r = 5 \)
    17. Center \(  (-4,5) \), radius  \(  r = \sqrt{42} \)
    19. Center  \(  \left( -\frac{5}{2}, \frac{1}{2}\right) \), radius  \(r = \frac{\sqrt{30}}{2} \)
    21. Center  \(  \left(-\frac{1}{2}, \frac{3}{5}\right) \), radius  \(r = \frac{\sqrt{161}}{10} \)
    23. This is not a circle.
      
    circleGraph11.png   circleGraph13.png
    circleGraph15.png circleGraph17.png circleGraph19.png circleGraph21.png

    D: Graph a circle given an equation in non-standard form

    Exercise \(\PageIndex{D}\)  

    \( \bigstar \) Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph it.

    1. \( x^2 + y^2 + 2x + 6y - 6  =0 \)
    2. \( x^2 + y^2 + 4x + 2y - 4  =0 \)
    3. \( x^2 + y^2 - 6x + 18y - 10  =0 \)
    4. \( x^2 + y^2 - 8x + 4y - 5  =0 \)
    5. \( x^2 + y^2 + 10x - 12y   = 13 \)
    6. \( x^2 + y^2 + 12x - 8y   = 5 \)
    1. \( x^2 + y^2 - 14x - 20y   = 20 \)
    2. \( x^2 + y^2 - 16x - 10y   = 11 \)
    3. \( x^2 + y^2 + x + 14y + \dfrac{53}{4}  =0 \)
    4. \( x^2 + y^2 +16 x + 5y = \tfrac{25}{4} \)
    5. \( x^2 + y^2  + 2y   = 3 \)
    6. \( x^2 + y^2 + 7x  -\tfrac{7}{4}  =0 \)
    Answers to Odd Numbered Problems:
    31. \( (x +1)^{2} + (y + 3)^{2} = 16 \),  Center: \((-1, -3)\), Radius: \(4\)
    33. \( (x - 3)^{2} + (y + 9)^{2} = 100 \),  Center: \((-3, 9)\), R: \(10\)  
    35. \( (x+5)^{2} + (y -6)^{2} = 74 \),  Center: \((-5, 6)\), R: \(\sqrt{74}\) 
    37. \( (x - 7)^{2} + (y -10)^{2} = 169 \),  Center: \((7, 10)\), Radius: \(13\)
    39. \( (x + 0.5)^{2} + (y + 7)^{2} = 36 \),  Center: \((-\tfrac{1}{2}, -7)\), Radius: \(6\) 
    41. \( x^{2} + (y + 1)^{2} = 4 \),  Center: \((0, -1)\), Radius = \(2\)
    circleGraph31.png 3.2e #33.png
    circleGraph35.png circleGraph37.png 3.2e #39.png circleGraph41.png

    E: Match equations with graphs of ellipses

    Exercise \(\PageIndex{E}\) 

    \( \bigstar \) Match each graph with one of the equations A–D.

    A. \(\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\) B. \(\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1\) C. \(\dfrac{x^2}{9} + {y^2} = 1\) D. \({x^2} + \dfrac{y^2}{9} = 1\)
    41. 屏幕快照 2019-07-30 上午12.53.34.png 42. 3.2e #42.png 43. 屏幕快照 2019-07-30 上午12.54.34.png 44.屏幕快照 2019-07-30 上午12.55.27.png

    \( \bigstar \) Match each graph to equations A-H.

    A. \(\dfrac{\left( {x - 2} \right)^2}{4} + \dfrac{{(y - 1)}^2}{9} = 1\)

    B. \(\dfrac{\left( {x - 2} \right)^2}{4} + \dfrac{{(y - 1)}^2}{16} = 1\)

    C. \(\dfrac{\left( {x - 2} \right)^2}{16} + \dfrac{{(y - 1)}^2}{4} = 1\)

    D. \(\dfrac{\left( {x - 2} \right)^2}{9} + \dfrac{{(y - 1)}^2}{4} = 1\)

    E. \(\dfrac{\left( {x + 2} \right)^2}{4} + \dfrac{{(y + 1)}^2}{9} = 1\)

    F. \(\dfrac{\left( {x + 2} \right)^2}{4} + \dfrac{{(y + 1)}^2}{16} = 1\)

    G. \(\dfrac{\left( {x + 2} \right)^2}{16} + \dfrac{{(y + 1)}^2}{4} = 1\)

    H. \(\dfrac{\left( {x + 2} \right)^2}{9} + \dfrac{{(y + 1)}^2}{4} = 1\)

    45.屏幕快照 2019-07-30 上午12.59.57.png 46.屏幕快照 2019-07-30 上午1.01.19.png 47. 屏幕快照 2019-07-30 上午1.02.03.png 49.屏幕快照 2019-07-30 上午1.02.44.png
    49.屏幕快照 2019-07-30 上午1.02.44.png 50. 屏幕快照 2019-07-30 上午1.03.28.png 51. 屏幕快照 2019-07-30 上午1.04.09.png 52.屏幕快照 2019-07-30 上午1.04.30.png
    Answers to Odd Numbered Problems:

    41. D      43. B      45. B      47. C      49. F      51. G  

    F: Write the equation for an ellipse given a graph

    Exercise \(\PageIndex{F}\) 

    \( \bigstar \) Write an equation for the graph.

    53. 屏幕快照 2019-07-30 上午12.56.57.png 54.屏幕快照 2019-07-30 上午12.57.36.png 55. 屏幕快照 2019-07-30 上午1.06.55.png 56.屏幕快照 2019-07-30 上午1.07.24.png

    \( \bigstar \) Find the standard form of the equation for an ellipse satisfying the given conditions.

    57. Center (0,0), horizontal radius = \(32 \), vertical radius = \(7 \)

    58. Center (0,0), horizontal radius = \(9 \), vertical radius = \(18 \)

    59. Center (0,0), horizontal radius = \( 2\), vertical radius = \( 3\)

    60. Center (0,0), horizontal radius = \(4 \), vertical radius = \(4 \)

    61. Center (-4, 3), horizontal radius = \(4 \), vertical radius = \(5\)

    62. Center (1, -2), horizontal radius = \(7 \), vertical radius = \(8 \)

    Answers to Odd Numbered Problems:

    53. \(\dfrac{x^2}{16} + \dfrac{y^2}{4} = 1\)

    55. \((x - 3)^2 + \dfrac{(y + 1)^2}{16} = 1\) 57. \(\dfrac{x^2}{1024} + \dfrac{y^2}{49} = 1\) 59. \(\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\)

    61. \(\dfrac{(x + 4)^2}{16} + \dfrac{(y - 3)^2}{25} = 1\)

    G: Graph an ellipse given an equation

    Exercise \(\PageIndex{G}\): Graph an Ellipse from an Equation in Standard Form

    \( \bigstar \) Find the center, and horizontal and vertical radii. Sketch the graph.

    63. \(\dfrac{x^2}{4} + \dfrac{y^2}{25} = 1\)

    64. \(\dfrac{x^2}{16} + \dfrac{y^2}{4} = 1\)

    65. \(\dfrac{x^2}{4} + y^2 = 1\)

    66. \(x^2 + \dfrac{y^2}{25} = 1\)

    67. \(x^2+ 25y^2 = 25\)

    68. \(16x^2 + y^2 = 16\)

    69. \(16x^2 + 9y^2 = 144\)

    70. \(16x^2 + 25y^2 = 400\)

    71. \(9x^2 + y^2 = 18\)

    72. \(x^2 + 4y^2 = 12\)

    Answers to Odd Numbered Problems:

    63. Center \((0, 0)\),
    Horiz. radius \(a = 2\),
    Vert. radius \(b = 5\)

    Screen Shot 2019-10-17 at 7.58.53 AM.png

    65. Center \((0, 0)\),
    Horiz. radius \(a = 2\),
    Vert. radius \(b = 1\)

    Screen Shot 2019-10-17 at 7.59.37 AM.png

    67. Center \((0, 0)\),
    Horiz. radius \(a = 5\),
    Vert. radius \(b = 1\)

    Screen Shot 2019-10-17 at 8.00.37 AM.png

    69. Center \((0, 0)\),
    Horiz. radius \(a = 3\),
    Vert. radius \(b = 4\)

    Screen Shot 2019-10-17 at 8.01.19 AM.png

    71. Center \((0, 0)\),
    Horiz. radius \(a = \sqrt{2}\),
    Vert. radius \(b =3\sqrt{2} \)

    Screen Shot 2019-10-17 at 8.02.06 AM.png

    H: Graph an ellipse given an equation in non-standard form

    Exercise \(\PageIndex{H}\) 

    \( \bigstar \) Find the center, and horizontal and vertical radii. Sketch the graph.

    73. \(\dfrac{(x - 1)^2}{25} + \dfrac{(y + 2)^2}{4} = 1\)

     

    74. \(\dfrac{(x + 5)^2}{16} + \dfrac{(y - 3)^2}{36} = 1\)  

    75. \((x + 2)^2 + \dfrac{(y - 3)^2}{25} = 1\)

     

    76. \(\dfrac{(x - 1)^2}{25} + (y - 6)^2 = 1\)

    77. \(4x^2 + 8x + 4 + y^2 = 16\)

    78. \(x^2 + 4y^2 + 16y + 16 = 36\)

    79. \(x^2 + 2x + 4y^2 + 16y = - 1\)

    80. \(4x^2 + 16x + y^2 - 8y = 4\)

    81. \(9x^2 - 36x + 4y^2 + 8y = 104\)

    82. \(4x^2 + 8x + 9y^2 + 36y = - 4\)

    Answers to Odd Numbered Problems:

    73. Center (1, -2), Horiz. radius \(a = 5\), Vert. radius \(b = 2\) 

    Screen Shot 2019-10-17 at 8.03.50 AM.png

    75. Center (-2, 3), Horiz. radius \(a = 1\), Vert. radius \(b = 5\)

    Screen Shot 2019-10-17 at 8.04.27 AM.png

    77. Center (-1, 0), Horiz. radius \(a = 2\), Vert. radius \(b = 4\)

    Screen Shot 2019-10-17 at 8.05.06 AM.png

    79. Center (-1, -2), Horiz. radius \(a = 4\), Vert. radius \(b = 2\)

    Screen Shot 2019-10-17 at 8.05.40 AM.png

    81. Center (2, -1), Horiz. radius \(a = 4\), Vert. radius \(b = 6\)

    Screen Shot 2019-10-17 at 8.06.09 AM.png


    3.2e: Circle Exercises. is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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