Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0) and B(6, -2), find M. A( 3, 7) and M(,-3), find B 5. M(, -) and B( -10, 11) find A 6. B(, ) and M(-2, 5), find A 7. Find the distance from A(, 2) to B(3, -).. Find the distance from A(5, 7) to B(-2, -).. 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(-, 7) to B(x, ) is 7, find x. M is the midpoint of A and B. Use the given information to find the missing point. 12. A(, -2) and B(5, 6), find M 13. A(, ) and B(-3, -7), find M 1. A(1, 10) and B(6, -2), find M 15. A(, ) and M(,-3), find B. M(, 7) and B( -10, 11) find A 17. B(-5, 10) and M(-2, 5), find A 1. Find the distance from A(-3, ) to B(3, -). 1. Find the distance from A(5, -) to B(-2, -). 20. Find the distance from A(-2,10) to B(-6, 0). 21. The distance from A(2, -3) to B(5, y) is 10, find y. 22. The distance from A(, 6) to B(2x, ) is 7, find x. Parabolas What is the vertex of the parabola? 23. y = (x 2) 2 + 2. y = 3(x + 5) 2 + 5. x = 5(y 7) 2 6 26. x = 2(y + ) 2 + 27. y = 2(x 7) 2 2. y = 3 (x)2 + 2. y = (x 7) 2 30. x = 5 3 (y + )2 3 Write the following equations in standard form. 31. y = x 2 + x 32. x = y 2 y 33. y = x 2 6x + Geometry - Conics ~1~ NJCTL.org
3. x = y 2 + 2y + 10 35. y = x 2 + 10x 12 36. x = y 2 y + 37. y = 2x 2 + 12x 3. x = 3y 2 6y 3. y = x 2 + x + 6 0. x = 6y 2 12y + 15 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. 1. y = 2(x ) 2 3 2. x = 3(y + 2) 2 6 3. y 2 (x + 6)2 + 5. x = 3 (y 5)2 + 7 5. y = (x 6) 2 6. x = 1 (y + 5)2 What is the vertex of the parabola? 7. y = (x + 3) 2 + 7. y = 2(x + ) 2 +. x = 6(y 3) 2 5 50. x = 2 3 (y + )2 10 51. y = (x 12) 2 11 52. y = 2(x 3) 2 53. y = (x) 2 + 5 5. x = 2 3 (y)2 Write the following equations in standard form. 55. y = x 2 + 6x 56. x = y 2 10y 57. y = x 2 x + 11 5. x = y 2 + y + 12 5. y = x 2 + x + 60. x = y 2 y + 61. y = 2x 2 + x 62. x = 3y 2 y 63. y = 5x 2 + 10x + 6. x = 2y 2 12y 30 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. 65. y = (x 2) 2 66. x = 5(y + 1) 2 7 67. y = 1 (x + )2 Geometry - Conics ~2~ NJCTL.org
6. x = 3 12 (y 2)2 + 1 6. y = (2x) 2 70. x = 3 (y + 6)2 Circles What are the center and the radius of the following circles? 71. (x + 2) 2 + (y ) 2 6 72. (x 3) 2 + (y 7) 2 = 73. (x) 2 + (y + ) 2 7. (x 7) 2 + (y + 1) 2 7 75. (x + 6) 2 + (y) 2 = 32 Write the standard form of the equation for the given information. 76. center (3,2) radius 6 77. center (-, -7) radius 7. center (5, -) radius 10 7. center (-, 0) diameter 1 0. center (,5) and point on the circle (3, -7) 1. diameter with endpoints (6, ) and (10, -) 2. center (, ) and tangent to the x-axis 3. x 2 + x + y 2 y 1. x 2 10x + y 2 + 2y 1 5. x 2 + 7x + y 2 1 What are the center and the radius of the following circles? 6. (x ) 2 + (y + 5) 2 = 7. (x + 11) 2 + (y ) 2 = 6. (x + 13) 2 + (y 3) 2. (x 2) 2 + (y) 2 0. (x 6) 2 + (y 15) 2 = 0 Write the standard form of the equation for the given information. 1. center (-2, -) radius 2. center (-3, 3) radius 11 3. center (5, ) radius 12. center (0, ) diameter 5. center (-,6) and point on the circle (-2, -) 6. diameter with endpoints (5, 1) and (11, -) 7. center (, ) and tangent to the y-axis. x 2 2x + y 2 + 10y 1. x 2 + 12x + y 2 + 20y 1 100. x 2 + x + y 2 y 2 Geometry - Conics ~3~ NJCTL.org
Ellipses Identify the ellipse s center and foci. State the length of the major and minor axes. Graph the ellipse. 101. 102. 103. 10. 105. (x 1) 2 + (y+3)2 + (y )2 1 (x) 2 + (y+5)2 36 (x+) 2 (x+1) 2 6 + (y+2)2 + (y 1)2 20 Write the equation of the ellipse in standard form with the following properties. 106. x 2 + x + 2y 2 y = 20 107. x 2 x + 3y 2 + 1y = 5 10. Center (1,), a horizontal major axis of 10 and a minor axis of 6. 10. Foci (2,5) and (2,11) with a minor axis of 10 110. Foci (-2,) and (-6,) with a major axis of 1 Identify the ellipse s center and foci. State the length of the major and minor axes. Graph the ellipse. 111. 112. 113. 11. 115. (x+5) 2 (x 7) 2 (x) 2 + (y)2 1 (x+1) 2 36 + (y )2 + (y+1)2 + (y)2 6 + (y 1)2 1 Write the equation of the ellipse in standard form with the following properties. 1. x 2 + 10x + 2y 2 12y = 1 117. 3x 2 12x + y 2 + y = 11. Center (-1,2), a vertical major axis of and a minor axis of. 11. Foci (3, 5) and (3,11) with a minor axis of 120. Foci (-2, 6) and (-, 6) with a major axis of 1 Hyperbolas Graph each of the following hyperbolas. Write the equations of the asymptotes. 121. 122. 123. (y+5) 2 (x 7) 2 (y 2) 2 (x )2 (y+1)2 (x)2 6 Geometry - Conics ~~ NJCTL.org
12. 1. (x) 2 (y)2 1 (y+1) 2 36 (x 1)2 1 Write the equation of the hyperbola in standard form. 126. x 2 + x 2y 2 y = 20 127. 3y 2 + 1y x 2 x 12. Opens horizontally, with center (3,7) and asymptotes with slope m = ± 2 5 12. Opens vertically, with asymptotes y = 3 2 x + and y = 3 2 x Graph each of the following hyperbolas. Write the equations of the asymptotes. 130. 131. 132. 133. 13. (y 1) 2 (y+3)2 (x )2 1 (x) 2 (y+5)2 36 (y+) 2 (y 6) 2 (x+2)2 (x+5)2 30 Write the equation of the hyperbola in standard form. 135. y 2 2y 5x 2 + 20x = 136. 6y 2 + 36y x 2 1x 137. Opens vertically, with center (-,1) and asymptotes with slope m = ± 3 7 13. Opens horizontally, with asymptotes y = x + 10 and y = x 1 Recognizing Conic Sections from the General Form Identify the conic section and state its eccentricity. Write the equation in standard form. 13. y 2 + 6y + x 2 + 10x 2 10. y 2 + y x 2 + 12x = 11. y 2 + y + 3x 2 1x = 7 12. y 2 + 2y x 2 + x = y 2 + 12 13. 2x 2 20x + 2y 2 + y = 6 1. x 2 2x 2y 2 + y = Identify the conic section and state its eccentricity. Write the equation in standard form. 15. y 2 + y + 2x 2 + 12x 0 16. y 2 + 2y x 2 + x 6 17. y 2 + y + x 2 2x 2 1. y 2 + 2y + x 2 + 12x = 2y 2 + 12 1. x 2 20x 2y 2 + y = 6 150. 6x 2 2x + y 2 + y = Geometry - Conics ~5~ NJCTL.org
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,) and M(5, -2). a. (,1) b. (,3) c. (7,-) d. (1,10) 3. What is the vertex of the parabola x = 2 3 (y )2 + 2 a. (,-2) b. (-2,2) c. (2,-2) d. (2,). Write the following equations in standard form x = 2y 2 + 12y + 2 a. x = 2(x + 6) 2 + 2 b. x = 2(x + 3) 2 7 c. x = 2(x + 3) 2 10 d. x = 2(x + 3) 2 5. Identify the focus of x = 2 (y 3)2 + 2 a. F(0,3) b. F(,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 (,-2) and focus (,). a. Directrix: y= -; Axis of Symmetry: x= b. Directrix: y= -10; Axis of Symmetry: x= c. Directrix: x= -; Axis of Symmetry: y= d. Directrix: x= -10; Axis of Symmetry: y= 7. Write the equation of the parabola with vertex (,-2) and focus (,). a. y (x )2 2 b. y (x )2 2 c. y 2 (x )2 2 d. x 12 (y + 2)2 + Geometry - Conics ~6~ NJCTL.org
. What are the center and the radius of the following circle: (x 7) 2 + (y + 6) 2 = a. (-7,6); r= b. (7,-6); r6 c. (-7,6); r= d. (7,-6); r= 2. Write the equation of the circle with a diameter with endpoints (6, 12) and (17, -). a. (x 11) 2 + (y 6) 2 = 521 b. (x 11) 2 + (y + 6) 2 = 22. c. (x 11) 2 + (y 2) 2 = 521 d. (x 11) 2 + (y 2) 2 = 22. 10. Identify the ellipse s center and foci: (x+)2 a. C(-,1); Foci: ( ± 20, 1) b. C(,-1); Foci: ( ± 20, 1) c. C(-,1); Foci: (,1 ± 20) d. C(,-1); Foci: (,1 ± 20) + (y 1)2 36 11. State the length of the major and minor axes of (x+)2 + (y 1)2 36 a. Major: ; Minor: 6 b. Major: 6; Minor: c. Major: 36; Minor: d. Major: 12; Minor: 12. Write the equation in standard form y 2 2y 2x 2 + 20x = 22 a. (y 3)2 2 b. (y 3)2 2 (y 3) 2 c. 27 d. (y 3)2 27 (x 5)2 (x+5)2 (x 5)2 5 (x+5)2 5 13. What is the slope of the asymptotes for the hyperbola (y+)2 a. y = ±2 b. y = ± 1 2 c. y = ± 2 2 d. y = ±2 2 1. Write the equation in standard form x 2 + 12x + 3y 2 12y = 1 a. (x + 6) 2 + 3(y 2) 2 = 7 b. (x+6)2 5 + (y 2)2 15 = 5 c. (x + 6) 2 + 3(y 2) 2 = 23 d. (x+6)2 23 + 3(y 2)2 23 = 7 (x+2)2 Geometry - Conics ~7~ NJCTL.org
15. Identify the conic section s eccentricity: y 2 y x 2 + 6x 2 a. e=0 b. 0<e<1 c. e d. e>1. Identify the conic section s eccentricity. y 2 + y + x 2 2x 2 a. e=0 b. 0<e<1 c. e d. e>1 Extended Response 1. A parabola has vertex (3, ) and focus (, ) a. What direction does the parabola open? b. What are the equations of the axis of symmetry and the directrix? c. Write the equation of the parabola. 2. Given the general form of a conic section as Ax 2 + Bx + Cy 2 + Dy + E = 0 a. What do A & C tell us about the conic? b. What is center of the conic if A 0 & C 0? c. If A, B, C, D are 2 and E is 0, what is eccentricity? 3. Consider a circle and a parabola. a. At how many points can they intersect? b. If the circle has equation x 2 + y 2 = and the parabola has equation y = x 2, what are the point(s) of intersection? c. If the parabola were reflected over the x-axis, what would be the point(s) of intersection? Geometry - Conics ~~ NJCTL.org
1. (3.5, -3) 2. (1.5, -1) 3. (,-1). (5,-13) 5. (1,-2) 6. (-,2) 7. 10.05. 17.6..7 10. -3 or 11. -+/-3 5 12. (.5, 2) 13. (3, -1.5) 1. (3.5, ) 15. (,-1). (26,3) 17. (1,0) 1. 1.03 1. 7 20. 10.77 21. -3 +/- 1 22. 2 +/ 10 23. (2,) 2. (-5,5). (-6,7) 26. (,-) 27. (7,-) 2. (0,) 2. (7,0) 30. (-3,-) 31. Y=(x+2) 2-32. S=(y-) 2-33. Y= (x-3) 2-1 3. (y+1) 2 +=x 35. Y=(x+5) 2-37 36. X= )y-) 2 37. Y=2(x+3) 2-1 Answers 3. X=3(y-1) 2-3 3. Y=-(x-1) 2 +10 0. X=-6(y+1) 2 +21 1. Up; v(,-3); F(,-2 7/); Dir: y=-3 1/; AOS x= 2. Left; v(-6,-2) F (-6 ½, -2); dir x=-5 11/12; AOS y=-2 3. Up v (-6, 5 ½); dir y= ½; AOS x=-6. right; v(7,5); F(7 1/3, 5) Dir x=6 2/3; AOS y=5 5. Down; V (6,-) F(6, - ¼) dir y=-7 3/ ; AOS x=6 6. Left; v (0,-5); F (-2,-5); dir x=2; AOS y=-5 7. (-3,7). (-,). (-5,3) 50. (-10,-) 51. (12,-11) 52. (3,0) 53. (0,-5) 5. (0,0) 55. Y=(x+3) 2-56. X= (Y-5) 2-57. Y= (x-2) 2 +7 5. X= (y+) 2-5. Y= (x+) 2-15 60. X=-(Y+) 2 +2 61. Y=2(x+2) 2-62. X=3 (y-1.5) 2-3.75 63. Y=-5(x-1) 2 +21 6. X=-2(Y+3) 2-12 65. Up v(2,-) F (2, -3 31/32) Dir y=- 1/32; AOS x=2 66. Left; V (-7,-1); F (-7 1/20, -1) Dir x= -6 1/20; AOS y=-1 67. Down; V (-,-); F (-,-); dir y=-7; AOS x=- Geometry - Conics ~~ NJCTL.org
6. Left; v (-1,2); F (-2,2); dir x=0; AOS y=2 6. Up; V (0,-); F (), -7 ¾); dir y=- ¼; AOS x=0 70. Right; V (0,-6); F (2/3, -6) Dir x=-2/3; AOS y=-6 71. C (-2,) r= 72. C (3,7); r=5 73. C (0,-); r 7. C (7,-1); r= 17 75. C (-6,0); r = 2 76. (x-3) 2 + (x-2) 2 =36 77. (x+) 2 + (Y+7) 2 =6 7. (x-5) 2 + (y+) 2 00 7. (x+) 2 + y 2 = 0. (x-) 2 + (y-5) 2 5 1. (x-) 2 + (y+2) 2 =0 2. (x-) 2 + (y-) 2 =1 3. (x+2) 2 + (y-) 2 =31. (x-5) 2 + (y+1) 2 =37 5. (x+3.5) 2 + y 2 =23. 6. C (,-5) r=3 7. C -11, ) r=. C(-13, 3) r2. C(2,0) r= 1 0. C (6,15) r=2 10 1. (x+2) 2 +(Y+) 2 =1 2. (x+3) 2 + (y-3) 2 21 3. (x-5) 2 + (y-) 2. X 2 + (y-) 2 =6 5. (x+) 2 + (y-6) 2 =200 6. (x-) 2 + (y-3) 2 30 7. (x-) 2 + ( y-) 2 6. (x-1) 2 + (y+5) 2 =37. (x+6) 2 + (y+10) 2 7 100. (x+2) 2 + (y-1) 2 = 101. C (2,-3); F1(2,-6.6) F2 (2,.6); maj=; min = 102. C (1,); F1 (3.3, ) F2(-1.3, ); maj=6; min=2 103. C (0,-5); f1 (0,-.32) F2 (0, -1.6); maj2; min0 10. C (-,-2); F1 (-6.3, -2) F2(-1.17, - 2); maj=; min = 2 105. C (-1,1); F1 (-1,.7); f2 (-1, 2.7); 106. 107. 10. 10. 110. maj = 5; min =2 6 (x+2) 2 32 (x 1) 2 (x 1) 2 (x+) 2 1 + (y 2)2 + (y+3)2 12 + (y )2 + (y+)2 3 + (y )2 37 111. C (-5,); F1(-7.65, ); F2 (-2.35, ); Maj= min =6 112. C (7,-1); F1 (&, 5.71); F2 (&,- 7.71); maj; min= 113. C (2,0); F1 (2, 6.); F2 (2, 6.); Maj6; min0 11. C(0,0); F1 (),3.); f1 (), -3.7); maj=; min=2 115. C(-1,1); F (3.23, 1); F2 (-5.2, 1); 1. 117. 11. 11. 120. maj 2; min =6 2 (x+5) 2 2 12 (x+1) 2 (x 3) 2 (x+5) 2 + (y 3)2 21 + (y+2)2 + (y 2)2 + (y )2 + (y 6)2 0 121. M = +/- /3 122. M= +/- 7/2 123. M= +/- 5/ 12. M= +/- 2 1. M= +/- 2 Geometry - Conics ~10~ NJCTL.org
126. 127. 12. 12. (x+2) 2 (y+3) 2 (x 3) 2 (y 2) 2 (y+2)2 (x+1)2 6 (y 7)2 + (x+)2 130. M = +/- 2 131. M= +/- 3 132. M+ +/- 6/5 133. M= +/- 2 13. M = +/- 30/10 135. 136. 137. 13. (y 3) 2 5 (y+3) 2 (x 2)2 (x+7)2 1 6 (y 1) 2 (x+)2 (x+27) 2 1 (y+2)2 13. Circle; x=0; (x+5) 2 + (Y+3) 2 =6 10. Hyperbola; e>1: (y+) 2 5 (x 6)2 5 11. Ellipse; o<e<1; 3(x 3) 2 50 + 2(y+2)2 12. Parabola; e; y/2(x-) 2-2 13. Circle; e=0; (x-5) 2 + (y+) 2 =3 1. Hyperbola; e>1; (x 3) 2 6 (y 2)2 12 15. Ellipse; x<e<; (x+3)2 + (y+1)2 16. Hyperbola; e>1 (Y+1) 2 (x-) 2 17. circle; e=0; (x-3)2+ (y+2)2 6 1. hyperbola; e>1; (x+6) 2 7 (y 1)2 7 1. hyperbola; e>1; (x 1) 2 62 (y )2 31 150. ellipse; 0<e<; + (y+1)2 6 Multiple Choice Answers 1. B 2. A 3. D. D 5. A 6. A 7. C. D. C 10. C 11. D 12. D 13. D 1. A 15. D. A Extended Response Answers 1. 2. 3. a. Right b. Axis of symmetry: y=, directrix: x=2 c. x (y )2 + 3 a. A and C identify the type of conic b. ( B 2A, D 2C ) c. e=0 a. 0, 1, or 2 points b. (-1.,1.56) and (1., 1.56) c. (-1.,-1.56) and (1., -1.56) Geometry - Conics ~11~ NJCTL.org