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3 ellipses

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3 ellipses

  1. 1. Ellipses
  2. 2. EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  3. 3. F2F1EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  4. 4. F2F1P QRIf P, Q, and R are anypoints on a ellipse,EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  5. 5. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  6. 6. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2= q1 + q2q1q2EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  7. 7. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2= q1 + q2= r1 + r2q1q2r2r1EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  8. 8. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2= q1 + q2= r1 + r2= a constantq1q2r2r1EllipsesGiven two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  9. 9. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2= q1 + q2= r1 + r2= a constantq1q2r2r1EllipsesAn ellipse has a center (h, k );(h, k)(h, k)Given two fixed points (called foci), an ellipse is the set ofpoints whose sum of the distances to the foci is a constant.
  10. 10. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2= q1 + q2= r1 + r2= a constantq1q2r2r1EllipsesAn 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 ofpoints whose sum of the distances to the foci is a constant.Major axisMajor axis
  11. 11. F2F1P QRp1p2If P, Q, and R are anypoints on a ellipse, thenp1 + p2= q1 + q2= r1 + r2= a constantq1q2r2r1EllipsesAn ellipse has a center (h, k ); it has two axes, the major(long) and the minor (short) axes.(h, k)Major axisMinor axis(h, k)Major axisMinor axisGiven two fixed points (called foci), an ellipse is the set ofpoints 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-radiusEllipsesx-radiusx-radius
  14. 14. y-radiusThese axes correspond to the important radii of the ellipse.From the center, the horizontal length is called the x-radiusand the vertical length the y-radius.Ellipsesx-radiusx-radiusy-radius
  15. 15. These axes correspond to the important radii of the ellipse.From the center, the horizontal length is called the x-radiusand the vertical length the y-radius.Ellipsesx-radiusThe general equation for ellipses isAx2+ By2+ Cx + Dy = Ewhere A and B are the same sign but different numbers.x-radiusy-radiusy-radius
  16. 16. These axes correspond to the important radii of the ellipse.From the center, the horizontal length is called the x-radiusand the vertical length the y-radius.Ellipsesx-radiusThe general equation for ellipses isAx2+ By2+ Cx + Dy = Ewhere A and B are the same sign but different numbers.Using completing the square, such equations may betransform to the standard form of ellipses below.x-radiusy-radiusy-radius
  17. 17. (x – h)2(y – k)2a2b2Ellipses+ = 1The Standard Form(of Ellipses)
  18. 18. (x – h)2(y – k)2a2b2Ellipses+ = 1 This has to be 1.The Standard Form(of Ellipses)
  19. 19. (x – h)2(y – k)2a2b2(h, k) is the center.Ellipses+ = 1 This has to be 1.The Standard Form(of Ellipses)
  20. 20. (x – h)2(y – k)2a2b2x-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)2a2b2x-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)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The Standard Form(of Ellipses)
  23. 23. (x – h)2(y – k)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The center is (3, –1).The Standard Form(of Ellipses)
  24. 24. (x – h)2(y – k)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The center is (3, –1).The x-radius is 4.The Standard Form(of Ellipses)
  25. 25. (x – h)2(y – k)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The 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)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The 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)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The 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)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The center is (3, –1).The x-radius is 4.The y-radius is 2.So the right point is (7, –1), the toppoint is (3, 1),The Standard Form(of Ellipses)
  29. 29. (x – h)2(y – k)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The center is (3, –1).The x-radius is 4.The y-radius is 2.So the right point is (7, –1), the toppoint is (3, 1), the left and bottompoints are (1, –1) and (3, –3).The Standard Form(of Ellipses)
  30. 30. (x – h)2(y – k)2a2b2x-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 andlabel the top, bottom, right and left most points.(x – 3)2(y + 1)24222+ = 1The center is (3, –1).The x-radius is 4.The y-radius is 2.So the right point is (7, –1), the toppoint is (3, 1), the left and bottompoints are (–1, –1) and (3, –3).The Standard Form(of Ellipses)
  31. 31. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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 thestandard 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 thestandard 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 = 11Ellipses
  34. 34. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11Ellipses
  35. 35. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the squareEllipses
  36. 36. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11Ellipses
  37. 37. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11+9Ellipses
  38. 38. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11+9 +16Ellipses
  39. 39. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16Ellipses
  40. 40. Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16Ellipses9(x – 1)2+ 4(y – 2)2= 36
  41. 41. 9(x – 1)24(y – 2)236 36Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16+ = 1Ellipses9(x – 1)2+ 4(y – 2)2= 36 divide by 36 to get 1
  42. 42. 9(x – 1)24(y – 2)236 4 36 9Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16+ = 1Ellipses9(x – 1)2+ 4(y – 2)2= 36 divide by 36 to get 1
  43. 43. 9(x – 1)24(y – 2)236 4 36 9Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16+ = 1(x – 1)2(y – 2)22232+ = 1Ellipses9(x – 1)2+ 4(y – 2)2= 36 divide by 36 to get 1
  44. 44. 9(x – 1)24(y – 2)236 4 36 9Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16+ = 1(x – 1)2(y – 2)22232+ = 1Ellipses9(x – 1)2+ 4(y – 2)2= 36 divide by 36 to get 1Hence, Center: (1, 2),x-radius is 2,y-radius is 3.
  45. 45. 9(x – 1)24(y – 2)236 4 36 9Example B. Put 9x2+ 4y2– 18x – 16y = 11 into thestandard 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-coefficients9(x2– 2x ) + 4(y2– 4y ) = 11 complete the square9(x2– 2x + 1 ) + 4(y2– 4y + 4 ) = 11 + 9 + 16+9 +16+ = 1(x – 1)2(y – 2)22232+ = 1Ellipses9(x – 1)2+ 4(y – 2)2= 36 divide by 36 to get 1Hence, Center: (1, 2),x-radius is 2,y-radius is 3.(-1, 2) (3, 2)(1, 5)(1, -1)(1, 2)
  46. 46. Ellipses

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