1. Ellipse
Conic Sections

2. Ellipse
The plane can intersect one nappe of the cone at an angle to the axis resulting in an 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| and c 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| and c is the distance from the center to the foci.

20. Ellipse - Graphing
where c2 = |a2 – b2| and c 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=| | c2 = 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=| | c2 = 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:
Distance to vertices in y direction: 5
Distance to foci:
c2=| | c2 = 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 = 13 Replace 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