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Ellipse

Ellipse. Conic Sections. The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse. Ellipse. Ellipse - Definition. An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant.

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Ellipse

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  1. Ellipse Conic Sections

  2. The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse. Ellipse

  3. Ellipse - Definition An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant. d1 + d2 = a constant value.

  4. Finding An Equation Ellipse

  5. Ellipse - Equation To find the equation of an ellipse, let the center be at (0, 0). The vertices on the axes are at (a, 0), (-a, 0),(0, b) and (0, -b). The foci are at (c, 0) and (-c, 0).

  6. Ellipse - Equation According to the definition. The sum of the distances from the foci to any point on the ellipse is a constant.

  7. Ellipse - Equation The distance from the foci to the point (a, 0) is 2a. Why?

  8. Ellipse - Equation The distance from (c, 0) to (a, 0) is the same as from (-a, 0) to (-c, 0).

  9. Ellipse - Equation The distance from (-c, 0) to (a, 0) added to the distance from (-a, 0) to (-c, 0) is the same as going from (-a, 0) to (a, 0) which is a distance of 2a.

  10. Ellipse - Equation Therefore, d1 + d2 = 2a. Using the distance formula,

  11. Ellipse - Equation Simplify: Square both sides. Subtract y2 and square binomials.

  12. Ellipse - Equation Simplify: Solve for the term with the square root. Square both sides.

  13. Ellipse - Equation Simplify: Get x terms, y terms, and other terms together.

  14. Ellipse - Equation Simplify: Divide both sides by a2(c2-a2)

  15. Ellipse - Equation Change the sign and run the negative through the denominator. At this point, let’s pause and investigate a2 – c2.

  16. Ellipse - Equation d1 + d2 must equal 2a. However, the triangle created is an isosceles triangle and d1 = d2. Therefore, d1 and d2 for the point (0, b) must both equal “a”.

  17. Ellipse - Equation This creates a right triangle with hypotenuse of length “a” and legs of length “b” and “c”. Using the pythagorean theorem, b2 + c2 = a2.

  18. Ellipse - Equation We now know….. and b2 + c2 = a2 b2 = a2 – c2 Substituting for a2 - c2 where c2 = |a2 – b2|

  19. Ellipse - Equation The equation of an ellipse centered at (0, 0) is …. where c2 = |a2 – b2|andc is the distance from the center to the foci. Shifting the graph over h units and up k units, the center is at (h, k) and the equation is where c2 = |a2 – b2| andc is the distance from the center to the foci.

  20. Ellipse - Graphing where c2 = |a2 – b2| andc is the distance from the center to the foci. Vertices are “a” units in the x direction and “b” units in the y direction. b a a c c The foci are “c” units in the direction of the longer (major) axis. b

  21. Graph - Example #1 Ellipse

  22. Ellipse - Graphing Graph: Center: (2, -3) Distance to vertices in x direction: 4 Distance to vertices in y direction: 5 Distance to foci: c2=|16 - 25| c2 = 9 c = 3

  23. Ellipse - Graphing Graph: Center: (2, -3) Distance to vertices in x direction: 4 Distance to vertices in y direction: 5 Distance to foci: c2=|16 - 25| c2 = 9 c = 3

  24. Graph - Example #2 Ellipse

  25. Ellipse - Graphing Graph: Complete the squares.

  26. Ellipse - Graphing Graph: Center: (-1, 3) Distance to vertices in x direction: 5 Distance to vertices in y direction: Distance to foci: c2=|25 - 10| c2 = 15 c =

  27. Find An Equation Ellipse

  28. Ellipse – Find An Equation Find an equation of an ellipse with foci at (-1, -3) and (5, -3). The minor axis has a length of 4. The center is the midpoint of the foci or (2, -3). The minor axis has a length of 4 and the vertices must be 2 units from the center. Start writing the equation.

  29. Ellipse – Find An Equation c2 = |a2 – b2|. Since the major axis is in the x direction, a2 > 4 9 = a2 – 4 a2 = 13Replace a2 in the equation.

  30. Ellipse – Find An Equation The equation is:

  31. Ellipse – Table Center: (h, k) Vertices: Foci: c2 = |a2 – b2| If a2 > b2 If b2 > a2

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