1. Conic Sections
The Ellipse Part A

2. Ellipse Another conic section formed by a plane intersecting a cone
Ellipse formed when

3. Definition of Ellipse Set of all points in the plane …
___________ of distances from two fixed points (foci) is a positive _____________

4. Definition of Ellipse
Definition demonstrated by using two tacks and a length of string to draw an ellipse

5. Note various parts of an ellipse
Graph of an Ellipse
Note various parts of an ellipse

6. Deriving the Formula
Note
Why?
Write with dist. formula
Simplify

7. Major Axis on y-Axis Standard form of equation becomes In both cases
Length of major axis = _______
Length of __________ axis = 2b

8. Using the Equation Given an ellipse with equation Determine foci
Determine values for a, b, and c
Sketch the graph

9. Find the Equation Given that an ellipse … What is the equation?
Has its center at (0,0)
Has a minor axis of length 6
Has foci at (0,4) and (0,-4)
What is the equation?

10. Ellipses with Center at (h,k)
When major axis parallel to x-axis equation can be shown to be

11. Ellipses with Center at (h,k)
When major axis parallel to y-axis equation can be shown to be

12. Find Vertices, Foci
Given the following equations, find the vertices and foci of these ellipses centered at (h, k)

13. Find the Equation Consider an ellipse with What is the equation?
Center at (0,3)
Minor axis of length 4
Focci at (0,0) and (0,6)
What is the equation?

14. Assignment
Ellipses A
1 – 43 Odd

15. Conic Sections
Ellipse The Sequel

16. Eccentricity A measure of the "roundness" of an ellipse not so round
very round

17. Eccentricity Given measurements of an ellipse Eccentricity
c = distance from center to focus
a = ½ the length of the major axis
Eccentricity

18. Eccentricity What limitations can we place on c in relationship to a?
_________________
What limitations does this put on
When e is close to 0, graph __________
When e close to 1, graph ____________

19. Finding the Eccentricity
Given an ellipse with
Center at (2,-2)
Vertex at (7,-2)
Focus at (4,-2)
What is the eccentricity?
Remember that

20. Using the Eccentricity
Consider an ellipse with e = ¾
Foci at (9,0) and (-9,0)
What is the equation of the ellipse in standard form?

21. Acoustic Property of Ellipse
Sound waves emanating from one focus will be reflected
Off the wall of the ellipse
Through the opposite focus

22. Whispering Gallery At Chicago Museum of Science and Industry
The Whispering Gallery is constructed in the form of an ellipsoid, with a parabolic dish at each focus. When a visitor stands at one dish and whispers, the line of sound emanating from this focus reflects directly to the dish/focus at the other end of the room, and to the other person!

23. Elliptical Orbits Planets travel in elliptical orbits around the sun
Or satellites around the earth

24. Elliptical Orbits Perihelion Aphelion Mean Distance
Distance from focus to ________________
Aphelion
Distance from _______ to farthest reach
Mean Distance
Half the ___________
Mean Dist

25. Elliptical Orbits
The mean distance of Mars from the Sun is 142 million miles.
Perihelion = million miles
Aphelion = ??
Equation for Mars orbit?
Mars

26. Assignment
Ellipses B
45 – 63 odd

27. Conic Sections
Ellipse Part 3

28. Additional Ellipse Elements
Recall that the parabola had a directrix
The ellipse has _________ directrices
They are related to the eccentricity
Distance from center to directrix =

29. Directrices of An Ellipse
An ellipse is the locus of points such that
The ratio of the distance to the nearer focus to …
The distance to the nearer directrix …
Equals a constant that is less than one.
This constant is the _______________.

30. Directrices of An Ellipse
Find the directrices of the ellipse defined by

31. Additional Ellipse Elements
The latus rectum is the distance across the ellipse ______________________
There is one at each focus.

32. Latus Rectum Consider the length of the latus rectum
Use the equation for an ellipse and solve for the y value when x = c
Then double that distance

33. Try It Out Given the ellipse What is the length of the latus rectum?
What are the lines that are the directrices?

34. Graphing An Ellipse On the TI
Given equation of an ellipse
We note that it is not a function
Must be graphed in two portions
Solve for y

35. Graphing An Ellipse On the TI
Use both results

36. Area of an Ellipse What might be the area of an ellipse?
If the area of a circle is …how might that relate to the area of the ellipse?
An ellipse is just a unit circle that has been stretched by a factor A in the x-direction, and a factor B in the y-direction

37. Area of an Ellipse
Thus we could conclude that the area of an ellipse is
Try it with
Check with a definite integral (use your calculator … it’s messy)

38. Assignment Ellipses C Exercises from handout 6.2
Also find areas of ellipse described in 73 and 79