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- 1. Ellipses
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. These axes correspond to the important radii of the ellipse. Ellipses
- 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. 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. 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. 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. (x – h)2 (y – k)2 a2 b2 Ellipses + = 1 The Standard Form (of Ellipses)
- 18. (x – h)2 (y – k)2 a2 b2 Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
- 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Ellipses
- 47. Ellipses
- 48. Ellipses (Answers)

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