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
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- Jasmin Reeves
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1 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(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1. 5, 1) 3. A( 2,0) and B(6, -2), find M. A( 3, 7) and M(,-3), find B (, 1) (5, 13) 5. M(, -) and B( -10, 11) find A 6. B(, 8) and M(-2, 5), find A (18, 2) ( 8, 2) 7. Find the distance from A(, 2) to B(3, -8). 8. 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. 2 5 y = 3 or 11. The distance from A(-, 7) to B(x, ) is 7, find x. x = ± or Pre-Calc Conics ~1~ NJCTL.org
2 Midpoint and Distance Formula Homework 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 (. 5, 2) (3, 1. 5) 1. A(1, 10) and B(6, -2), find M 15. A(, 8) and M(,-3), find B (3. 5, ) (, 1). M(8, 7) and B( -10, 11) find A 17. B(-5, 10) and M(-2, 5), find A (26, 3) (1, 0) 18. Find the distance from A(-3, ) to B(3, -8). 1. Find the distance from A(5, -) to B(-2, -) 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. 2 2 y = 3 ± or The distance from A(, 6) to B(2x, ) is 7, find x. x = 2 ± or 1. Pre-Calc Conics ~2~ NJCTL.org
3 Parabolas Class Work What is the vertex of the parabola? 23. (x 2) 2 = (y ) 2. (x + 5) 2 = 1 (y 5) 25. (y 3 7)2 (x + 6) 5 (2, ) ( 5, 5) ( 6, 7) Write the following equations in standard form. State the direction of the opening. Identify vertex and the focus and give the equations of the directrix and axis of symmetry. 26. x 2 + x y = y 2 8y x = 0 (x + 2) 2 = (y + ) V: ( 2, ) F: ( 2, 3. 75) D: y =. 25 A: x = 2 (y ) 2 = (x + ) V: (, ) F: ( , ) D: x =. 25 A: y = 28. x 2 6x y + 8 = 0 2. y 2 + 2y x + 10 = 0 (x 3) 2 = (y + 1) V: (3, 1) F: (3, 0. 75) D: y = A: x = 3 (y + 1) 2 = (x ) V: (, 1) F: (. 25, 1) D: x = A: y = x x y 12 = y 2 8y x + = 0 (x + 5) 2 = (y + 37) V: ( 5, 37) F: ( 5, ) D: y = A: x = 5 (y ) 2 = x V: (0, ) F: (0. 25, ) D: x = A: y = 32. 2x x 10y 2 = y 2 6y 12x + 3 = 0 (x + 3) 2 = 5(y + 2) V: ( 3, 2) F: ( 3, 0. 8) D: y = 3. 2 A: x = 3 (y 1) 2 = (x 3) V: (3, 1) F: (, 1) D: x = 2 A: y 3. x 2 + 8x 8y 100 = y 2 + 8y 36x + 8 = 0 (x 1) 2 = 12(y + 2) V: (1, 2) F: (1, 5) D: y A: x (y ) 2 = 6(x ) V: (, ) F: (2. 5, ) D: x = 5. 5 A: y = Pre-Calc Conics ~3~ NJCTL.org
4 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. 36. (x + 6) 2 = 2(y 5) 37. (y + 5) 2 = 8x V: ( 6, 5) F: ( 6, 5. 5) D: y =. 5 A: x = 6 V: (0, 5) F: ( 2, 5) D: x = 2 A: y = (x 3) 2 = (y + 1) 3. (y ) 2 2(x + 2) V: (3, 1) F: (3, 5) D: y = 3 A: x = 3 V: ( 2, ) F: (1, ) D: x = 5 A: y = 0. y 2 + 2y + 12x 23 = 0 1. x 2 2x y + = 0 (y + 1) 2 2(x 2) V: (2, 1) F: ( 1, 1) D: x = 5 A: y = 1 (x 1) 2 = (y 2) V: (1, 2) F: (1, 3) D: y A: x Pre-Calc Conics ~~ NJCTL.org
5 Parabolas Homework What is the vertex of the parabola? 2. (x + 3) 2 = (y 7) 3. (x + ) 2 = 1 (y 8). (y 2 3)2 (x + 5) 6 (2, ) (, 8) ( 5, 3) Write the following equations in standard form. State the direction of the opening. Identify vertex and the focus and give the equations of the directrix and axis of symmetry. 5. x 2 + 6x y = 0 6. y 2 10y x = 0 (x + 3) 2 = (y + ) V: ( 3, ) F: ( 3, 8. 75) D: y =. 25 A: x = 3 (y 5) 2 = (x + 25) V: ( 25, 5) F: ( 2. 75, 5) D: x = A: y = 5 7. x 2 x y + 11 = 0 8. y 2 + 8y x + 12 = 0 (x 2) 2 = (y 7) V: (2, 7) F: (2, 7. 25) D: y = A: x = 2 (y + ) 2 = (x + ) V: (, ) F: ( 3. 75, ) D: x =. 25 A: y =. x 2 + x y + = y 2 8y x + 8 = 0 (x + 8) 2 = (y + 15) V: ( 8, 15) F: ( 8, 1. 75) D: y = A: x = 8 (y + ) 2 = (x 2) V: (2, ) F: (23. 75, ) D: x = A: y = 51. 2x 2 + 8x y = y 2 18y 2x 5 = 0 (x + 2) 2 = 2(y + 2) V: ( 2, 2) F: ( 2, 1. 5) D: y = 2. 5 A: x = 2 (y 3) 2 = 8(x + 3) V: ( 3, 3) F: ( 1, 3) D: x = 5 A: y = x x 20y + 15 = y 2 12y 20x 78 = 0 (x 1) 2 = (y 1) V: (1, 1) F: (1, 0) D: y = 2 A: x (y + 3) 2 = 10(x + 3) V: ( 3, 3) F: ( 5. 5, 3) D: x = 0. 5 A: y = 3 Pre-Calc Conics ~5~ NJCTL.org
6 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. 55. (x 2) 2 = 8(y + ) 56. (y + 1) 2 = (x + 7) V: (2, ) F: (2, 2) D: y = 6 A: x = 2 V: ( 7, 1) F: ( 11, 1) D: x = 3 A: y = (x + ) 2 = (y + 8) 58. y 2 2y + 8x = 0 V: (, 8) F: (, ) D: y = 7 A: x = (y 12) 2 = 8(x 1) V: (1, 12) F: ( 1, 12) D: x = 3 A: y 2 5. x 2 y 8 = x 2 + 2x 12y + 2 = 0 x 2 = (y + 2) V: (0, 2) F: (0, 1) D: y = 3 A: x = 0 (x + ) 2 = (y + 2) V: (, 2) F: (, 1) D: y = 3 A: x = Pre-Calc Conics ~6~ NJCTL.org
7 Circles Class Work What are the center and the radius of the following circles? 61. (x + 2) 2 + (y ) (x 3) 2 + (y 7) 2 = (x) 2 + (y + 8) 2 C: ( 2, ) r: C: (3, 7) r: 5 C: (0, 8) r: 1 6. (x 7) 2 + (y + 1) (x + 6) 2 + (y) 2 = 32 C: (7, 1) r: 17 C: ( 6, 0) r: 2 Write the standard form of the equation for the given information. 66. center (3,2) radius center (-, -7) radius center (5, -) radius 10 (x 3) 2 + (y 2) 2 = 36 (x + ) 2 + (y + 7) 2 = 6 (x 5) 2 + (y + ) center (-8, 0) diameter center (,5) and point on the circle (3, -7) (x + 8) 2 + y 2 = (x ) 2 + (y 5) diameter with endpoints (6, ) and (10, -8) 72. center (, ) and tangent to the x-axis (x 8) 2 + (y + 2) 2 = 0 (x ) 2 + (y ) 2 = 81 Write the standard form of the equation, identify the Center and Radius, then graph. 73. x 2 + x + y 2 8y = 5 7. x 2 10x + y 2 + 2y x x + y 2 3 (x + 2) 2 + (y ) 2 = 25 C: ( 2, ) r: 5 (x 5) 2 + (y + 1) 2 = 36 C: (5, 1) r: 6 (x + 6) 2 + y 2 = C: ( 6, 0) r: 7 Pre-Calc Conics ~7~ NJCTL.org
8 Circles Homework What are the center and the radius of the following circles? 76. (x ) 2 + (y + 5) 2 = 77. (x + 11) 2 + (y 8) 2 = (x + 13) 2 + (y 3) 2 C: (, 5) r: 3 C: ( 11, 8) r: 8 C: ( 13, 3) r: (x 2) 2 + (y) (x 6) 2 + (y 15) 2 = 0 C: (2, 0) r: 1 C: (6, 15) r: 2 10 Write the standard form of the equation for the given information. 81. center (-2, -) radius 82. center (-3, 3) radius center (5, 8) radius 12 (x + 2) 2 + (y + ) 2 = 81 (x + 3) 2 + (y 3) 2 21 (x 5) 2 + (y 8) 2 8. center (0, 8) diameter 85. center (-,6) and point on the circle (-2, -8) x 2 + (y 8) 2 = 6 (x + ) 2 + (y 6) 2 = diameter with endpoints (5, 1) and (11, -8) 87. center (, ) and tangent to the y-axis (x 8) 2 + (y 3) 2 30 (x ) 2 + (y ) 2 6 Write the standard form of the equation, identify the Center and Radius, then graph. 88. x 2 2x + y y 0 8. x x + y y = 8 0. x 2 + x + y 2 8y 6 (x 1) 2 + (y + 5) 2 = 36 C: (1, 5) r: 6 (x + 6) 2 + (y + 10) 2 C: ( 6, 10) r: 12 (x + 2) 2 + (y 1) 2 = C: ( 2, 1) r: 3 Pre-Calc Conics ~8~ NJCTL.org
9 Ellipses Class Work State whether the ellipse is vertical or horizontal, and the length of the major and minor axes. Identify the ellipse s center, vertices, and foci. Graph the ellipse. 1. (x 2)2 + (y+3)2 2. (x 1)2 + (y )2 1 C: (2, 3) Vertical Major: 8 Minor: V: (2, 1)(2, 7)(0, 3) (, 3) F: (2, 0. 6)(2, 6. 6) C: (1, ) Horizontal Major: 6 Minor: 2 V: (, )( 2, )(1, 5) (1, 3) F: (3. 83, )( 1. 83, ) 3. (x)2 + (y+5) (x+)2 + (y+2)2 8 C: (0, 5) Vertical Major: 12 Minor: 10 V: (0, 1)(0, 11)(5, 5) ( 5, 5) F: (0, 1. 68)(0, 8. 32) C: (, 2) Horizontal Major: 8 Minor: V: (0, 2)( 8, 2)(, 0. 83) (,. 83) F: ( 1. 17, 2)( 6. 83, 2) 5. (x+1)2 + (y 1) (x 3) (y+6)2 C: ( 1, 1) Vertical Major: 8. Minor:. V: ( 1, 5. 7)( 1, 3. 7)(1. 5, 1) ( 3. 5, 1) F: ( 1,. 7)( 1, 2. 7) C: (3, 6) Horizontal Major: 10 Minor: 6 V: (8, 6)( 2, 6)(3, 3) (3, ) F: (7, 6)( 1, 6) Pre-Calc Conics ~~ NJCTL.org
10 Write the equation of the ellipse in standard form with the following properties. 7. x 2 + x + 2y 2 8y = x 2 8x + 3y y = 5 (x+2) (y 2)2 (x 2) (y+3)2. Center (1,), a horizontal major axis of 10 and a minor axis of 6. (x 1) (y ) Foci (2,5) and (2,11) with a minor axis of Foci (-2,) and (-6,) with a major axis of 18 (x 2) (y 8)2 3 (x+) (y )2 77 Pre-Calc Conics ~10~ NJCTL.org
11 Ellipses Homework State whether the ellipse is vertical or horizontal, and the length of the major and minor axes. Identify the ellipse s center, vertices, and foci. Graph the ellipse (x+5) 2 + (y ) (x 7)2 + (y+1)2 C: ( 5, ) Horizontal Major: 8 Minor: 6 V: ( 1, )(, )( 5, 7) ( 5, 1) F: ( 2. 35, )( 7. 65, ) C: (7, 1) Vertical Major: 1 Minor: V: (7, 6)(7, 8)(, 1) (5, 1) F: (7, 5. 71)(7, 7. 71) 10. (x 2) (y) (x)2 1 + (y)2 C: (2, 0) Vertical Major: Minor: 10 V: (2, 8)(2, 8)(7, 0) ( 3, 0) F: (2, 6. 25)(2, 6. 25) C: (0, 0) Vertical Major: 8 Minor: 1 V: (0, 2)(0, 2)(1, 0) ( 1, 0) F: (0, 1. 73)(0, 1. 73) 106. (x+1) (y 1) (x+3)2 + (y 5)2 25 C: ( 1, 1) Horizontal Major: 12 Minor: 8. V: (5, 1)( 7, 1)( 1, 5. 2) ( 1, 3. 2) F: ( 5. 2, 1)(3. 2, 1) C: ( 3, 5) Horizontal Major: 26 Minor: 10 V: (10, 5)(, 5)( 3, 10) ( 3, 0) F: (, 5)( 15, 5) Pre-Calc Conics ~11~ NJCTL.org
12 Write the equation of the ellipse in standard form with the following properties x x + 2y 2 12y = x 2 12x + y 2 + y = 8 (x+5) (y 3)2 21 (x 2) (y+2) Center (-1,2), a vertical major axis of 8 and a minor axis of. (x+1) 2 + (y 2) Foci (3, 5) and (3,11) with a minor axis of Foci (-2, 6) and (-8, 6) with a major axis of 1 (x 3) 2 + (y 8)2 25 (x+5) 2 + (y 6)2 0 Pre-Calc Conics ~12~ NJCTL.org
13 Hyperbolas Class Work State whether the hyperbola is vertical or horizontal, identify the center, vertices, foci, and the slopes of the asymptotes. Graph the hyperbola (y+5) 2 (x )2 Vert C: (, 5) Asy: ± 3 V: (, 1)(, ) F: (, 0)(, 10) 11. (x 7)2 (y+1)2 Horiz C: (7, 1) Asy: ± 7 2 V: (, 1)(5, 1) F: (1. 28, 1)( 0. 28, 1) 115. (y 2)2 25 (x)2 6 Vert C: (0, 2) Asy: ± 5 8 V: (0, 7)(0, 3) F: (0, 11. 3)(0, 7. 3) 1. (x) 2 (y) (y+1)2 36 (x 1)2 18 Horiz C: (0, 0) Asy: ±2 V: (1, 0)( 1, 0) F: (5. 2, 0)( 5. 2, 0) Vert C: (1, 1) Asy: ± 2 V: (1, 5)(1, 7) F: (1, 6. 35)(1, 8. 35) Write the equation of the hyperbola in standard form x 2 + x 2y 2 8y = y y x 2 8x (x+2) 2 (y+2)2 8 (y+3) 2 8 (x+1) Opens horizontally, with center (3,7) and asymptotes with slope m = ± 2 5 (x 3) Opens vertically, with asymptotes y = 3 2 x + 8 and y = 3 2 x (y 2) 2 (y 7)2 (x+)2 Pre-Calc Conics ~13~ NJCTL.org
14 Hyperbolas Homework State whether the hyperbola is vertical or horizontal, identify the center, vertices, foci, and the slopes of the asymptotes. Graph the hyperbola (x 2) 2 (y+3)2 Horiz C: (2, 3) Asy: ±2 V: (, 3)(0, 3) F: (6. 7, 3)( 2. 7, 3) 123. (y 1)2 (x )2 1 Vert C: (, 1) Asy: ±3 V: (, )(, 2) F: (,. )(, 2. ) 12. (x)2 (y+5) Horiz C: (0, 5) Asy: ± 6 5 V: (5, 5)( 5, 5) F: (7. 81, 5)( 7. 81, 5) 125. (y+) 2 (x+2) (y 6)2 (x+5)2 30 Vert C: ( 2, ) Asy: ± 2 V: ( 2, 0)( 2, 8) F: ( 2, 0. )( 2, 8. ) Vert C: ( 5, 6) Asy: ± V: ( 5, )( 5, 3) F: ( 5, )( 5, 0. 25) Write the equation of the hyperbola in standard form y 2 2y 5x x = y y x 2 1x (y 3) 2 5 (x 2)2 (y+3) 2 1 (x+7) Opens vertically, with center (-,1) and asymptotes with slope m = ± 3 7 (y 1) Opens horizontally, with asymptotes y = x + 10 and y = x 1 (x+27) 2 (x+)2 (y+2)2 81 Pre-Calc Conics ~1~ NJCTL.org
15 Recognizing Conic Sections from the General Form Class Work Identify the conic section and write the equation in standard form. State all pertinent information y 2 + 6y + x x y 2 + 8y x x = 2 Circle (y + 3) 2 + (x + 5) 2 = C: ( 5, 3) r = 7 Vertical Hyperbola (y+) 2 (x 6)2 C: (6, ) V: (6, 2)(6, 6) F: (6, 1. 17)(6, 6. 83) Asy: ± y 2 + y + 3x 2 18x = y 2 + 2y x 2 + 8x = y Horizontal Ellipse (y+2) (x 3)2 C: (3, 2) Major: 8 Minor: 6. 2 V: (7, 2)( 1, 2) (3, 1. 6)(3, 5. 6) F: (5, 2)(1, 2) Vertical Parabola (x 2) 2 = 2(y ) V: (2, ) F: (2, 3. 5) D: y =. 5 A: x = x 2 20x + 2y 2 + y = x 2 2x 2y 2 + 8y = Circle (x 5) 2 + (y + ) 2 = 36 C: (5, ) r = 6 Horizontal Hyperbola (x 3) 2 6 (y 2)2 12 C: (3, 2) V: (5. 5, 2)(0. 55, 2) F: (7. 2, 2)( 1. 2, 2) Asy: ± 2 Pre-Calc Conics ~15~ NJCTL.org
16 Recognizing Conic Sections from the General Form Homework Identify the conic section and write the equation in standard form. State all pertinent information y 2 + 8y + 2x x y 2 + 2y x 2 + 8x 6 Horizontal Ellipse (y+1) (x+3)2 C: ( 3, 1) Major: 8 Minor: V: (1, 1)( 7, 1) ( 3, 1. 83)( 3, 3. 83) F: (5, 2)(1, 2) Vertical Hyperbola (y+1) 2 1 (x )2 1 C: (, 1) V: (, 0)(, 2) F: (, 0. 1)(, 2. 1) Asy: ±1 13. y 2 + y + x 2 2x y 2 + 2y + x x = 2y Circle (y + 2) 2 + (x 3) 2 6 C: (3, 2) r = Horizontal Parabola (y + 1) 2 = 12(x 1) V: (1, 1) F: ( 2, 1) D: x = A: y = x 2 20x 2y 2 + y = x 2 2x + y 2 + 8y = Horizontal Hyperbola (x 10) 2 6 (y )2 32 C: (10, ) V: (18, )(2, ) F: (1. 8, )(0. 2, ) Asy: ± 2 2 Vertical Ellipse (x 2) 2 + (y+1)2 6 C: (2, 1) Major:. Minor: V: (2, 1. 5)(2, 3. 5) (, 1)(0, 1) F: (2, 0. 1)(2, 2. 1) Pre-Calc Conics ~~ NJCTL.org
17 Unit Review - Multiple Choice 1. The distance from A(2,y) to B(-1,7) is 5. Find y. a. 3 b. D c. 11 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,-8) C d. (1,10) 3. What is the vertex of the parabola (y ) 2 = (x 2) a. (,-2) b. (-2,2) D c. (2,-2) d. (2,). Write the following equations in standard form 2y y x + 2 = 0 a. (y + 6) 2 (x 2) 2 b. (y + 3) 2 (x + 7) 2 D c. (y + 3) 2 (x + 10) 2 d. (y + 3) 2 (x + ) 2 5. Identify the focus of (y 3) 2 8(x 2) a. F(0,3) b. F(,3) A 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= -8; Axis of Symmetry: x= b. Directrix: y= -10; Axis of Symmetry: x= A c. Directrix: x= -8; 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. (x ) 2 6(y + 2) b. (x ) 2 = 8(y + 2) c. (x ) 2 = 2(y + 2) C d. (y + 2) 2 2(x ) 8. What are the center and the radius of the following circle: (x 7) 2 + (y + 6) 2 = a. (-7,6); r= b. (7,-6); r= D c. (-7,6); r= 8 d. (7,-6); r= 2. Write the equation of the circle with a diameter with endpoints (6, 12) and (, -8). a. (x 11) 2 + (y 6) 2 25 b. (x 11) 2 + (y + 6) C c. (x 11) 2 + (y 2) 2 25 d. (x 11) 2 + (y 2) Pre-Calc Conics ~17~ NJCTL.org
18 10. Identify the ellipse s center and foci: (x+)2 a. C(-,1); Foci: ( ± 5, 1) b. C(,-1); Foci: ( ± 5, 1) c. C(-,1); Foci: (,1 ± 5) d. C(,-1); Foci: (,1 ± 5) + (y 1) State the length of the major and minor axes of (x+)2 + (y 1)2 36 a. Major: ; Minor: 6 b. Major: 6; Minor: D c. Major: 36; Minor: d. Major: 12; Minor: Write the equation in standard form y 2 2y 2x x = 22 a. b. c. d. (y 3) 2 2 (y 3) 2 2 (y 3) 2 27 (y 3) 2 27 (x 5)2 (x+5)2 (x 5)2 5 (x+5) What is the slope of the asymptotes for the hyperbola (y+)2 a. y = ±2 b. y = ± 1 2 D c. y = ± 2 2 d. y = ± 2 1. Write the equation in standard form x x + 3y 2 12y = 3 a. (x + 6) 2 + 3(y 2) 2 = 5 b. (x+6) (y 2)2 15 c. (x + 6) 2 + 3(y 2) 2 = 23 d. (x+6) (y 2) Identify the type of conic section: y 2 y x 2 + 6x 2 a. Circle b. Ellipse D c. Hyperbola d. Parabola. Identify the type of conic section. y 2 + y + x 2 2x 2 a. Hyperbola b. Circle B c. Parabola d. Ellipse C A B (x+2)2 8 Pre-Calc Conics ~18~ NJCTL.org
19 Short Answer Identify the conic section, graph, and write in standard form. State all pertinent information: (Parabolas direction, vertex, focus, directrix, axis of symmetry; Circles center, radius; Ellipse direction, center, vertices, foci, major axis, minor axis; Hyperbola direction, center, vertices, foci, slope of asymptotes) 1. x 2 6x + y 2 32y = y 2 6y x + 5 = 0 Circle (x 3) 2 + (y ) 2 C: (3, ) r Parabola - Right (y 3) 2 = (x + 1) V: ( 1, 3) F: ( 3, 3) D: x = 1 1 A: y = 3 3. y 2 32y x 2 18x 37. x 2 2x + y 2 882y = 380 Hyperbola - Vertical (y 1) 2 (x+1)2 C: (1, 1) V: ( 1, )( 1, 2) F: ( 1, 6)( 1, ) A: ± 3 Ellipse - Horizontal (x 3) 2 + (y )2 C: (3, ) V: (10, )(, )(3, 7)(3, 11) F: (10. 28, )(. 28, ) Major: 1 Minor: 5. x 2 6x + 8y 31 = x 2 126x + y 2 + 6y = 352 Parabola - Down (x 3) 2 = 8(y 5) V: (3, 5) F: (3, 3) D: y = 7 A: x = 3 Ellipse - Vertical (x ) 2 + (y+2)2 81 C: (, 2) V: (, 7)(, 11)(8, 2)(0, 2) F: (, 7. 85)(, ) Major: 18 Minor: 8 7. x 2 + 2x + y y = 0 8. x 2 25y y = 200 Circle (x + 12) 2 + (y + 5) 2 = C: ( 12, 5) r = 3 Hyperbola - Horizontal x 2 (y 2)2 25 C: (0, 2) V: (5, 2)( 5, 2) F: (5., 2)( 5., 2) A: ± 2 5 Pre-Calc Conics ~1~ NJCTL.org
20 Extended Response 1. A parabola has vertex (3, ) and focus (, ) a. What direction does the parabola open? Right b. What are the equations of the axis of symmetry and the directrix? D: x = 2 A: y = c. Write the equation of the parabola. (y ) 2 = (x 3) 2. Consider a circle and a parabola. a. At how many points can they intersect? 0, 1, 2, 3, or 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? (1. 25, 1. 56) and ( 1. 25, 1. 56) c. If the parabola were reflected over the x-axis, what would be the point(s) of intersection? (1. 25, 1. 56) and ( 1. 25, 1. 56) Pre-Calc Conics ~20~ NJCTL.org
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
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)
Conic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.
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)
Circles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).
Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.
Distance and Midpoint Formula 7.1
Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units
The Distance Formula. The Midpoint Formula
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