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ellipse

ellipse. By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org. Last Updated: March 11, 2008. Ellipse. The set of all co-planar points whose sum of the distances from two fixed points (foci) are constant. Ellipse.

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ellipse

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  1. ellipse By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: March 11, 2008

  2. Ellipse • The set of all co-planar points whose sum of the distances from two fixed points (foci) are constant.

  3. Ellipse • The set of all points whose sum of the distances from two fixed points (foci) are constant. Look at an animation

  4. Ellipse • Definition of major and minor axis minor axis major axis

  5. Ellipse major axis minor axis

  6. Ellipse • Definition of vertices • and co-vertices co-verticies verticies verticies endpoint ofmajor axis co-verticies endpoint ofminor axis

  7. Ellipse Distance from center to vertex = a Distance from center to co-vertex = b Distance from center to foci = c a b c a Length of major axis = 2a Length of minor axis = 2b a2 = b2 + c2

  8. Ellipse a b c a Distance from center to vertex = a a2 = b2 + c2 Distance from center to co-vertex = b Distance from center to foci = c

  9. Ellipse a a c a2 = b2 + c2 b Distance from center to vertex = a Distance from center to co-vertex = b Distance from center to foci = c

  10. Ellipse The Latus Rectum (LR) is a chord passing through the focus that is perpendicular to the major axis. The length of the L.R. is Distance from center to vertex = a Distance from center to co-vertex = b Distance from center to foci = c

  11. Graph the following Ellipse Center: (2, -3) a = 5 in x direction b = 3 in y direction (2, 0) 3 (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6)

  12. Graph the following Ellipse a = 5 b = 3 a2 = b2 + c2 52 = 32 + c2 25 = 9 + c2 foci 16 = c2 4 = c (2, 0) 3 (-2, -3) (6, -3) (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6)

  13. Graph the following Ellipse a = 5 b = 3 c = 4 Find the Length of the LR. (2, 0) 3 (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6)

  14. Graph the following Ellipse Center: (2, -3) Major Axis: length = 10 endpoints (-3, -3) & (7, -3) Minor Axis: length = 6 endpoints (2, 0) & (2, -6) Foci: (-2, -3) (6, -3) (2, 0) Length of LR: 3 (-2, -3) (6, -3) (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6)

  15. Graph the following Ellipse x + 1 = 0 y - 5 = 0 x = -1 y = 5 (-1, 12) Center: (-1, 5) 7 a = 7 in y direction (-1, 5) (-5, 5) (3, 5) 4 4 b = 4 in x direction 7 (-1, -2)

  16. Graph the following Ellipse a = 7 b = 4 a2 = b2 + c2 72 = 42 + c2 49 = 16 + c2 (-1, 12) 33 = c2 foci 7 (-1, 5) (-5, 5) (3, 5) 4 4 7 (-1, -2)

  17. Graph the following Ellipse Find theLength of the LR. a = 7 b = 4 c = (-1, 12) 7 (-1, 5) (-5, 5) (3, 5) 4 4 7 (-1, -2)

  18. Graph the following Ellipse Center: (-1, 5) Major Axis: length = 14 endpoints (-1, -2) & (1, 12) (-1, 12) Minor Axis: length = 8 endpoints (-5, 5) & (3, 5) 7 (-1, 5) (-5, 5) (3, 5) 4 4 Foci: 7 Length of LR: (-1, -2)

  19. Graph the following Ellipse 4x2 + 8x + 9y2 + 54y + 52 = 3 (4x2 + 8x ) + (9y2 + 54y ) = 3 - 52 4(x2 + 2x + 12) + 9(y2 + 6y + 32) = -49 + 4 + 81 4(x + 1)2 + 9(y + 3)2 = 36 36 9 4

  20. Graph the following Ellipse Center: (-1, -3) a = 3 in x direction b = 2 in y direction (-1, -1) 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5)

  21. Graph the following Ellipse a = 3 b = 2 a2 = b2 + c2 32 = 22 + c2 9 = 4 + c2 foci 5 = c2 (-1, -1) 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5)

  22. Graph the following Ellipse a = 3 b = 2 c = Find the Length of the LR. (-1, -1) 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5)

  23. Graph the following Ellipse Center: (-1, -3) Major Axis: length = 6 endpoints (-4, -3) & (2, -3) Minor Axis: length = 4 endpoints (-1, -1) & (-1, -5) Foci: (-1, -1) Length of LR: 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5)

  24. Graph the following Ellipse 9x2 + 36x + 4y2 - 40y - 100 = 88 (9x2 + 36x ) + (4y2 - 40y ) = 88 + 100 9(x2 + 4x + 22) + 4(y2 - 10y + (-5)2) = 188 + 36 + 100 9(x + 2)2 + 4(y -5)2 = 324 324 36 81

  25. Graph the following Ellipse Center: (-2, 5) a = 9 in y direction b = 6 in x direction (-2, 14) 9 (-2, 5) (-8, 5) (4, 5) 6 6 9 (-2, -4)

  26. Graph the following Ellipse a = 9 b = 6 a2 = b2 + c2 92 = 62 + c2 81 = 36 + c2 (-2, 14) 45 = c2 foci 9 (-2, 5) (-8, 5) (4, 5) 6 6 9 (-2, -4)

  27. Graph the following Ellipse Find theLength of the LR. a = 9 b = 6 c = (-2, 14) 9 (-2, 5) (-8, 5) (4, 5) 6 6 9 (-2, -4)

  28. Graph the following Ellipse Center: (-2, 5) (-2, 14) Major Axis: length = 18 endpoints (-2, 14) & (-2, -4) 9 Minor Axis: length = 12 endpoints (-8, 5) & (4, 5) (-2, 5) (-8, 5) (4, 5) 6 6 9 Foci: Length of LR: (-2, -4)

  29. Graph the following Ellipse 4x2 - 32x + 25y2 - 150y + 189 = 0 (4x2 – 8x ) + (25y2 - 150y ) = 0 - 189 4(x2 - 8x + (-4)2) + 25(y2 - 6y + (-3)2) = -189 + 64 + 225 4(x - 4)2 + 25(y - 3)2 = 100 100 25 4

  30. Graph the following Ellipse Center: (4, 3) a = 5 in x direction b = 2 in y direction (4, 5) 2 (4, 3) (-1, 3) (9, 3) 5 5 2 (4, 1)

  31. Graph the following Ellipse a = 5 b = 2 a2 = b2 + c2 52 = 22 + c2 25 = 4 + c2 foci 21 = c2 (4, 5) 2 (4, 3) (-1, 3) (9, 3) 5 5 2 (4, 1)

  32. Graph the following Ellipse a = 5 b = 2 c = Find the Length of the LR. (4, 5) 2 (4, 3) (-1, 3) (9, 3) 5 5 2 (4, 1)

  33. Graph the following Ellipse Center: (4, 3) Major Axis: length = 10 endpoints (-1, 3) & (9, 3) Minor Axis: length = 4 endpoints (4, 5) & (4, 1) Foci: (4, 5) Length of LR: 2 (4, 3) (-1, 3) (9, 3) 5 5 2 (4, 1)

  34. Graph the following Ellipse 7x2 + 28x + 5y2 - 2y + 15 = 17 (7x2 + 28x ) + (5y2 - 2y ) = 17 - 15 7(x2 + 4x + 22) + 5(y2 - 2y + (-1)2) = 2 + 28 + 5 7(x + 2)2 + 5(y - 1)2 = 35 35 5 7

  35. Graph the following Ellipse Center: (1, -2) a = in y direction b = in x direction (1, -2)

  36. Graph the following Ellipse a = b = a2 = b2 + c2 foci (1, -2)

  37. Graph the following Ellipse Length of the LR. (1, -2)

  38. Graph the following Ellipse Center: (1, -2) Major Axis: length = endpoints Minor Axis: length = endpoints (1, -2) Foci: Length of LR:

  39. Elliptical Billiard table • http://cage.ugent.be/~hs/billiards/billiards.html • http://www.cut-the-knot.org/Curriculum/Geometry/EnvelopesInEllipse.shtml

  40. That's All Folks

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