3.2e: Circle Exercises.

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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.

$\left(x-2\right)^{2} + \left(y+3\right)^{2} = 9$
$\left(x+1\right)^{2} + \left(y-2\right)^{2} = 16$
$ (x - 2)^{2} + (y + 5)^{2} = 4 $
$ (x + 1)^{2} + (y + 5)^{2} = 100 $
$ (x + 9)^{2} + y^{2} = 25 $
$ (x-4)^2+(y+2)^2 = 9 $
$ (x+4)^2 + (y-5)^2 = 42 $

$ (x-3)^2+(y-6)^2 = 20 $
$ \left(x + \frac{5}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 = \frac{30}{4} $
$ (x-1)^2 + (y-5)^2 = 5 $
$ \left(x + \frac{1}{2}\right)^{2} + \left(y - \frac{3}{5}\right)^{2} = \frac{161}{100} $
$ (x - 3)^{2} + (y - 5)^{2} = 65 $
$ x^{2} + (y - 3)^{2} = 0 $
$ 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.

$ x^2 + y^2 + 2x + 6y - 6 =0 $
$ x^2 + y^2 + 4x + 2y - 4 =0 $
$ x^2 + y^2 - 6x + 18y - 10 =0 $
$ x^2 + y^2 - 8x + 4y - 5 =0 $
$ x^2 + y^2 + 10x - 12y = 13 $
$ x^2 + y^2 + 12x - 8y = 5 $

$ x^2 + y^2 - 14x - 20y = 20 $
$ x^2 + y^2 - 16x - 10y = 11 $
$ x^2 + y^2 + x + 14y + \dfrac{53}{4} =0 $
$ x^2 + y^2 +16 x + 5y = \tfrac{25}{4} $
$ x^2 + y^2 + 2y = 3 $
$ 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. $屏幕快照 2019-07-30 上午1.02.44.png$

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 $b = 6$

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

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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.		$circleGraph11.png$	$circleGraph13.png$
$circleGraph15.png$	$circleGraph17.png$	$circleGraph19.png$	$circleGraph21.png$

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 = \(2\)		$circleGraph31.png$	$3.2e #33.png$
$circleGraph35.png$	$circleGraph37.png$	$3.2e #39.png$	$circleGraph41.png$

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. $屏幕快照 2019-07-30 上午12.55.27.png$