Conic Sections

Midpoint and Distance Formula

Class Work

M is the midpoint of A and B. Use the given information to find the missing point.

  1. A(4, 2) and B(3, -8), find M
  2. A(5, 7) and B( -2, -9), find M
  3. A( 2,0) and B(6, -2), find M
  4. A( 3, 7) and M(4,-3), find B
  5. M(4, -9) and B( -10, 11) find A
  6. B(4, 8) and M(-2, 5), find A
  7. Find the distance from A(4, 2) to B(3, -8).
  8. Find the distance from A(5, 7) to B(-2, -9).
  9. Find the distance from A(2,0) to B(6, -2).
  10. The distance from A(2, 3) to B(-6, y) is 10, find y.
  11. The distance from A(-4, 7) to B(x, 9) is 7, find x.

Homework

M is the midpoint of A and B. Use the given information to find the missing point.

  1. A(4, -2) and B(5, 6), find M
  2. A(9, 4) and B(-3, -7), find M
  3. A(1, 10) and B(6, -2), find M
  4. A( 4, 8) and M(4,-3), find B
  5. M(8, 7) and B( -10, 11) find A
  6. B(-5, 10) and M(-2, 5), find A
  7. Find the distance from A(-3, 9) to B(3, -8).
  8. Find the distance from A(5, -9) to B(-2, -9).
  9. Find the distance from A(-2,10) to B(-6, 0).
  10. The distance from A(2, -3) to B(5, y) is 10, find y.
  11. The distance from A(4, 6) to B(2x, 9) is 7, find x.

Parabolas

Class Work

What is the vertex of the parabola?

Write the following equations in standard form.

Graph each of the following. State the direction of the opening. Identify vertex and the focus and give the equations of the directrix and axis of symmetry.

Homework

What is the vertex of the parabola?

Write the following equations in standard form.

Graph each of the following. State the direction of the opening. Identify vertex and the focus and give the equations of the directrix and axis of symmetry.

Circles

Class Work

What are the center and the radius of the following circles?

Write the standard form of the equation for the given information.

  1. center (3,2) radius 6
  2. center (-4, -7) radius 8
  3. center (5, -9) radius 10
  4. center (-8, 0) diameter 14
  5. center (4,5) and point on the circle (3, -7)
  6. diameter with endpoints (6, 4) and (10, -8)
  7. center (4, 9) and tangent to the x-axis

Homework

What are the center and the radius of the following circles?

Write the standard form of the equation for the given information.

  1. center (-2, -4) radius 9
  2. center (-3, 3) radius 11
  3. center (5, 8) radius 12
  4. center (0 , 8) diameter 16
  5. center (-4,6) and point on the circle (-2, -8)
  6. diameter with endpoints (5, 14) and (11, -8)
  7. center (4, 9) and tangent to the y-axis

Ellipses

Class Work

Identify the ellipse’s center and foci. State the length of the major and minor axes. Graph the ellipse.

Write the equation of the ellipse in standard form with the following properties.

  1. Center (1,4), a horizontal major axis of 10 and a minor axis of 6.
  2. Foci (2,5) and (2,11) with a minor axis of 10
  3. Foci (-2,4) and (-6,4) with a major axis of 18

Homework

Identify the ellipse’s center and foci. State the length of the major and minor axes. Graph the ellipse.

Write the equation of the ellipse in standard form with the following properties.

  1. Center (-1,2), a vertical major axis of 8 and a minor axis of 4.
  2. Foci (3, 5) and (3,11) with a minor axis of 8
  3. Foci (-2, 6) and (-8, 6) with a major axis of 14

Hyperbolas

Class Work

Graph each of the following hyperbolas. Write the equations of the asymptotes.

Write the equation of the hyperbola in standard form.

  1. Opens horizontally, with center (3,7) and asymptotes with slope
  2. Opens vertically, with asymptotes and

Homework

Graph each of the following hyperbolas. Write the equations of the asymptotes.

Write the equation of the hyperbola in standard form.

  1. Opens vertically, with center (-4,1) and asymptotes with slope
  2. Opens horizontally, with asymptotes and

Recognizing Conic Sections from the General Form

Class Work

Identify the conic section and state its eccentricity. Write the equation in standard form.

Homework

Identify the conic section and state its eccentricity. Write the equation in standard form.

Multiple Choice

1. The distance from A(2,y) to B(-1,7) is 5. Find y.

a. 2

b. 3

c. 12

d. A and C

2. M is the midpoint of EF. Find F given E(3,4) and M(5, -2).

a. (4,1)

b. (4,3)

c. (7,-8)

d. (1,10)

3. What is the vertex of the parabola

a. (9,-2)

b. (-2,2)

c. (2,-2)

d. (2,9)

4. Write the following equations in standard form

a.

b.

c.

d.

5. Identify the focus of

a. F(0,3)

b. F(4,3)

c. F(2,1)

d. F(2,5)

6. Write the equations of the directrix and axis of symmetry of a parabola with vertex (4,-2) and focus (4,4).

a. Directrix: y= -8; Axis of Symmetry: x=4

b. Directrix: y= -10; Axis of Symmetry: x=4

c. Directrix: x= -8; Axis of Symmetry: y=4

d. Directrix: x= -10; Axis of Symmetry: y=4

7. Write the equation of the parabola with vertex (4,-2) and focus (4,4).

a.

b.

c.

d.

8. What are the center and the radius of the following circle:

a. (-7,6); r=4

b. (7,-6); r=16

c. (-7,6); r= 8

d. (7,-6); r= 2

9. Write the equation of the circle with a diameter with endpoints (6, 12) and (17, -8).

a.

b.

c.

d.

10. Identify the ellipse’s center and foci:

a. C(-4,1); Foci:

b. C(4,-1); Foci:

c. C(-4,1); Foci:

d. C(4,-1); Foci:

11. State the length of the major and minor axes of

a. Major: 4; Minor: 6

b. Major: 6; Minor: 4

c. Major: 36; Minor: 16

d. Major: 12; Minor: 8

12. Write the equation in standard form

a.

b.

c.

d.

13. What is the slope of the asymptotes for the hyperbola

a.

b.

c.

d.

14. Write the equation in standard form

a.

b.

c.

d.

15. Identify the conic section’s eccentricity:

a. e=0

b. 0<e<1

c. e=1

d. e>1

16. Identify the conic section’s eccentricity.

a. e=0

b. 0<e<1

c. e=1

d. e>1

Extended Response

1. A parabola has vertex (3, 4) and focus (4, 4)

  1. What direction does the parabola open?
  2. What are the equations of the axis of symmetry and the directrix?
  3. Write the equation of the parabola.

2. Given the general form of a conic section as

  1. What do A & C tell us about the conic?
  2. What is center of the conic if
  3. If A, B, C, D are 2 and E is 0, what is eccentricity?

3. Consider a circle and a parabola.

  1. At how many points can they intersect?
  2. If the circle has equation and the parabola has equation , what are the point(s) of intersection?
  3. If the parabola were reflected over the x-axis, what would be the point(s) of intersection?

Answers

Geometry - Conics~1~NJCTL.org

  1. (3.5, -3)
  2. (1.5, -1)
  3. (4,-1)
  4. (5,-13)
  5. (18,-29)
  6. (-8,2)
  7. 10.05
  8. 17.46
  9. 4.47
  10. -3 or 9
  11. -4+/-
  12. (4.5, 2)
  13. (3, -1.5)
  14. (3.5, 4)
  15. (4,-14)
  16. (26,3)
  17. (1,0)
  18. 18.03
  19. 7
  20. 10.77
  21. -3 +/-
  22. (2,4)
  23. (-5,5)
  24. (-6,7)
  25. (9,-4)
  26. (7,-9)
  27. (0,4)
  28. (7,0)
  29. (-3,-8)
  30. Y=(x+2)2 -4
  31. S=(y-4)2 -16
  32. Y= (x-3)2 -1
  33. (y+1)2+9=x
  34. Y=(x+5)2 -37
  35. X= )y-4)2
  36. Y=2(x+3)2-18
  37. X=3(y-1)2 -3
  38. Y=-4(x-1)2 +10
  39. X=-6(y+1)2 +21
  40. Up; v(4,-3); F(4,-2 7/8); Dir: y=-3 1/8; AOS x=4
  41. Left; v(-6,-2) F (-6 ½, -2); dir x=-5 11/12; AOS y=-2
  42. Up v (-6, 5 ½); dir y=4 ½; AOS x=-6
  43. right; v(7,5); F(7 1/3, 5) Dir x=6 2/3; AOS y=5
  44. Down; V (6,-8) F(6, -8 ¼) dir y=-7 3/4 ; AOS x=6
  45. Left; v (0,-5); F (-2,-5); dir x=2; AOS y=-5
  46. (-3,7)
  47. (-4,8)
  48. (-5,3)
  49. (-10,-8)
  50. (12,-11)
  51. (3,0)
  52. (0,-5)
  53. (0,0)
  54. Y=(x+3)2 -9
  55. X= (Y-5)2 -25
  56. Y= (x-2)2 +7
  57. X= (y+4)2 -4
  58. Y= (x+8)2 -15
  59. X=-(Y+4)2 +24
  60. Y=2(x+2)2 -8
  61. X=3 (y-1.5)2 -3.75
  62. Y=-5(x-1)2 +21
  63. X=-2(Y+3)2 -12
  64. Up v(2,-4) F (2, -3 31/32) Dir y=-4 1/32; AOS x=2
  65. Left; V (-7,-1); F (-7 1/20, -1) Dir x= -6 19/20; AOS y=-1
  66. Down; V (-9,-8); F (-9,-9); dir y=-7; AOS x=-9
  67. Left; v (-1,2); F (-2,2); dir x=0; AOS y=2
  68. Up; V (0,-8); F (), -7 ¾); dir y=-8 ¼; AOS x=0
  69. Right; V (0,-6); F (2/3, -6) Dir x=-2/3; AOS y=-6
  70. C (-2,4) r=4
  71. C (3,7); r=5
  72. C (0,-8); r=1
  73. C (7,-1); r=
  74. C (-6,0); r =4
  75. (x-3)2 + (x-2)2 =36
  76. (x+4)2 + (Y+7)2 =64
  77. (x-5)2 + (y+9)2 = 100
  78. (x+8)2 + y2 =49
  79. (x-4)2 + (y-5)2 =145
  80. (x-8)2 + (y+2)2 =40
  81. (x-4)2 + (y-9)2 =81
  82. (x+2)2 + (y-4)2 =31
  83. (x-5)2 + (y+1)2 =37
  84. (x+3.5)2 + y2 =23.25
  85. C (9,-5) r=3
  86. C -11, 8) r=8
  87. C(-13, 3) r=12
  88. C(2,0) r=
  89. C (6,15) r=2
  90. (x+2)2 +(Y+4)2 =81
  91. (x+3)2 + (y-3)2 =121
  92. (x-5)2 + (y-8)2 =144
  93. X2 + (y-8)2 =64
  94. (x+4)2 + (y-6)2 =200
  95. (x-8)2 + (y-3)2 = 130
  96. (x-4)2 + ( y-9)2 =16
  97. (x-1)2 + (y+5)2 =37
  98. (x+6)2 + (y+10)2 =147
  99. (x+2)2 + (y-1)2 =8
  100. C (2,-3); F1(2,-6.46) F2 (2, .46); maj=8; min =4
  101. C (1,4); F1 (3.83, 4) F2(-1.83, 4); maj=6; min=2
  102. C (0,-5); f1 (0,-8.32) F2 (0, -1.68); maj=12; min=10
  103. C (-4,-2); F1 (-6.83, -2) F2(-1.17, -2); maj=8; min =4
  104. C (-1,1); F1 (-1,4.74); f2 (-1, 2.74); maj =4; min =2
  105. =1
  106. =1
  107. =1
  108. =1
  109. =1
  110. C (-5,4); F1(-7.65, 4); F2 (-2.35, 4); Maj=8 min =6
  111. C (7,-1); F1 (&, 5.71); F2 (&,-7.71); maj=14; min=4
  112. C (2,0); F1 (2, 6.25); F2 (2, 6.25); Maj=16; min=10
  113. C(0,0); F1 (),3.8); f1 (), -3.87); maj=4; min=2
  114. C(-1,1); F (3.23, 1); F2 (-5.24, 1); maj =12; min =6
  115. =1
  116. =1
  117. =1
  118. =1
  119. =1
  120. M = +/- 4/3
  121. M= +/- 7/2
  122. M= +/- 5/8
  123. M= +/- 2
  124. M= +/-
  125. =1
  126. =1
  127. =1
  128. =1
  129. M = +/- 2
  130. M= +/- 3
  131. M+ +/- 6/5
  132. M= +/-
  133. M = +/- /10
  134. =1
  135. =1
  136. =1
  137. =1
  138. Circle; x=0; (x+5)2 + (Y+3)2=46
  139. Hyperbola; e>1:

=1

  1. Ellipse; o<e<1;

=1

  1. Parabola; e=1; y=1/2(x-4)2 -2
  2. Circle; e=0; (x-5)2 + (y+4)2 =38
  3. Hyperbola; e>1;

=1

  1. Ellipse; x<e<4; =1
  2. Hyperbola; e>1 (Y+1)2 –(x-4)2=1
  3. circle; e=0; (x-3)2+ (y+2)2 =16
  4. hyperbola; e>1;

=1

  1. hyperbola; e>1;

=1

  1. ellipse; 0<e<4;

=1

Multiple Choice Answers

  1. B
  2. A
  3. D
  4. D
  5. A
  6. A
  7. C
  8. D
  9. C
  10. C
  11. D
  12. D
  13. D
  14. A
  15. D
  16. A

Extended Response Answers

  1. Right
  2. Axis of symmetry: y=4,

directrix: x=2

  1. A and C identify the type of conic
  2. e=0
  3. 0, 1, or 2 points
  4. (-1.25,1.56) and (1.25, 1.56)
  5. (-1.25,-1.56) and (1.25, -1.56)

Geometry - Conics~1~NJCTL.org