3.4 ellipses

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3.4 ellipses

  1. 1. Ellipses
  2. 2. Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  3. 3. F2F1 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  4. 4. F2F1 P Q R If P, Q, and R are any points on a ellipse, Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  5. 5. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  6. 6. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 q1 q2 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  7. 7. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 q1 q2 r2r1 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  8. 8. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  9. 9. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses An ellipse has a center (h, k ); (h, k) (h, k) Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  10. 10. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses An ellipse has a center (h, k ); it has two axes, the major (long) (h, k) (h, k) Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Major axis Major axis
  11. 11. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses An ellipse has a center (h, k ); it has two axes, the major (long) and the minor (short) axes. (h, k)Major axis Minor axis (h, k) Major axis Minor axis Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
  12. 12. These axes correspond to the important radii of the ellipse. Ellipses
  13. 13. These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius Ellipses x-radius x-radius
  14. 14. y-radius These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius. Ellipses x-radius x-radius y-radius
  15. 15. These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius. Ellipses x-radius The general equation for ellipses is Ax2 + By2 + Cx + Dy = E where A and B are the same sign but different numbers. x-radius y-radiusy-radius
  16. 16. These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius. Ellipses x-radius The general equation for ellipses is Ax2 + By2 + Cx + Dy = E where A and B are the same sign but different numbers. Using completing the square, such equations may be transform to the standard form of ellipses below. x-radius y-radiusy-radius
  17. 17. (x – h)2 (y – k)2 a2 b2 Ellipses + = 1 The Standard Form (of Ellipses)
  18. 18. (x – h)2 (y – k)2 a2 b2 Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
  19. 19. (x – h)2 (y – k)2 a2 b2 (h, k) is the center. Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
  20. 20. (x – h)2 (y – k)2 a2 b2 x-radius = a (h, k) is the center. Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
  21. 21. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
  22. 22. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The Standard Form (of Ellipses)
  23. 23. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The Standard Form (of Ellipses)
  24. 24. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The Standard Form (of Ellipses)
  25. 25. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. The Standard Form (of Ellipses)
  26. 26. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. The Standard Form (of Ellipses)
  27. 27. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), The Standard Form (of Ellipses)
  28. 28. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1) (3, 1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), The Standard Form (of Ellipses)
  29. 29. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1)(-1, -1) (3, -3) (3, 1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), the left and bottom points are (1, –1) and (3, –3). The Standard Form (of Ellipses)
  30. 30. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1)(-1, -1) (3, -3) (3, 1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), the left and bottom points are (–1, –1) and (3, –3). The Standard Form (of Ellipses)
  31. 31. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Ellipses
  32. 32. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: Ellipses
  33. 33. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 Ellipses
  34. 34. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 Ellipses
  35. 35. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square Ellipses
  36. 36. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 Ellipses
  37. 37. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 +9 Ellipses
  38. 38. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 +9 +16 Ellipses
  39. 39. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 Ellipses
  40. 40. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36
  41. 41. 9(x – 1)2 4(y – 2)2 36 36 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
  42. 42. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
  43. 43. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 (x – 1)2 (y – 2)2 22 32 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
  44. 44. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 (x – 1)2 (y – 2)2 22 32 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1 Hence, Center: (1, 2), x-radius is 2, y-radius is 3.
  45. 45. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 (x – 1)2 (y – 2)2 22 32 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1 Hence, Center: (1, 2), x-radius is 2, y-radius is 3. (-1, 2) (3, 2) (1, 5) (1, -1) (1, 2)
  46. 46. Ellipses
  47. 47. Ellipses
  48. 48. Ellipses (Answers)

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