1. 10.2 The Ellipse
Day Two
Galatians 2:20 "I have been crucified with Christ. It is no
longer I who live, but Christ who lives in me. And the life I
now live in the flesh I live by faith in the Son of God, who
loved me and gave himself for me."

2. The eccentricity of an ellipse is how much it
varies from being a circle ... it is the ratio of
c to a
c 2 2
e= where c = a − b
a

3. The eccentricity of an ellipse is how much it
varies from being a circle ... it is the ratio of
c to a
c 2 2
e= where c = a − b
a
0 < e <1

4. The eccentricity of an ellipse is how much it
varies from being a circle ... it is the ratio of
c to a
c 2 2
e= where c = a − b
a
0 < e <1
e close to 0 is very circular
e close to 1 is really stretched out

5. 1. The vertices of an ellipse are ( ± 6,0 ) and
the foci are ( ± 4,0 ) . Find its equation.

6. 1. The vertices of an ellipse are ( ± 6,0 ) and
the foci are ( ± 4,0 ) . Find its equation.
2 2 2
a=6 c=4 c = a −b

7. 1. The vertices of an ellipse are ( ± 6,0 ) and
the foci are ( ± 4,0 ) . Find its equation.
2 2 2
a=6 c=4 c = a −b
2 2 2
4 = 6 −b

8. 1. The vertices of an ellipse are ( ± 6,0 ) and
the foci are ( ± 4,0 ) . Find its equation.
2 2 2
a=6 c=4 c = a −b
2 2 2
4 = 6 −b
2 2 2
b =6 −4

9. 1. The vertices of an ellipse are ( ± 6,0 ) and
the foci are ( ± 4,0 ) . Find its equation.
2 2 2
a=6 c=4 c = a −b
2 2 2
4 = 6 −b
2 2 2
b =6 −4
2
b = 20

10. 1. The vertices of an ellipse are ( ± 6,0 ) and
the foci are ( ± 4,0 ) . Find its equation.
2 2 2
a=6 c=4 c = a −b
2 2 2
4 = 6 −b
2 2 2
b =6 −4
2
b = 20
2 2
x y
+ =1
36 20

11. 2. Find the foci of the ellipse 9x + 4y = 36
2 2

12. 2. Find the foci of the ellipse 9x + 4y = 36
2 2
2 2
x y
+ =1
4 9

13. 2. Find the foci of the ellipse 9x + 4y = 36
2 2
2 2
x y
+ =1
4 9
2 2 2
c = a −b

14. 2. Find the foci of the ellipse 9x + 4y = 36
2 2
2 2
x y
+ =1
4 9
2 2 2
c = a −b
2
c = 9−4 = 5

15. 2. Find the foci of the ellipse 9x + 4y = 36
2 2
2 2
x y
+ =1
4 9
2 2 2
c = a −b
2
c = 9−4 = 5
c=± 5

16. 2. Find the foci of the ellipse 9x + 4y = 36
2 2
2 2
x y
+ =1
4 9
2 2 2
c = a −b
2
c = 9−4 = 5
c=± 5
(
F 0, ± 5 )

17. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5

18. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
c = 20

19. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
c = 20
c
e=
a

20. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
c = 20
c
e=
a
4 20
=
5 a

21. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
c = 20
c
e=
a
4 20
=
5 a
a = 25

22. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
c = 20
c
e=
a
4 20
=
5 a
a = 25
2
a = 625

23. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
2
c = 20 c = 400
c
e=
a
4 20
=
5 a
a = 25
2
a = 625

24. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
2
c = 20 c = 400
c
e= 2 2
c = a −b 2
a
4 20
=
5 a
a = 25
2
a = 625

25. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
2
c = 20 c = 400
c
e= 2 2
c = a −b 2
a
2 2 2
4 20 b =a −c
=
5 a
a = 25
2
a = 625

26. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
2
c = 20 c = 400
c
e= 2 2
c = a −b 2
a
2 2 2
4 20 b =a −c
= 2
5 a b = 625 − 400
a = 25
2
a = 625

27. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =
5
2
c = 20 c = 400
c
e= 2 2
c = a −b 2
a
2 2 2
4 20 b =a −c
= 2
5 a b = 625 − 400
a = 25 2
b = 225
2
a = 625

28. 3. Find the equation of the ellipse with foci
4
( 0, ± 20 ) and eccentricity e =5
2
c = 20 c = 400
c
e= 2 2
c = a −b 2
a
2 2 2
4 20 b =a −c
= 2
5 a b = 625 − 400
a = 25 2
b = 225
2
a = 625 2
x y 2
+ =1
225 625

29. 4. Find the vertices, foci and eccentricity of
the ellipse, 4x + y = 16 . Determine the
2 2
lengths of the major and minor axes and
sketch the graph.

30. 4. Find the vertices, foci and eccentricity of
the ellipse, 4x + y = 16 . Determine the
2 2
lengths of the major and minor axes and
sketch the graph.
2 2
x y
+ =1
4 16

31. 4. Find the vertices, foci and eccentricity of
the ellipse, 4x + y = 16 . Determine the
2 2
lengths of the major and minor axes and
sketch the graph.
2 2
x y
+ =1
4 16
a=4 b=2
2
c = 16 − 4 = 12
c=2 3

32. 4. Find the vertices, foci and eccentricity of
the ellipse, 4x + y = 16 . Determine the
2 2
lengths of the major and minor axes and
sketch the graph.
x2
y 2 vertices : ( 0, ± 4 )
+ =1
4 16 foci : ( 0, ± 2 3 )
a=4 b=2 3
e:
2
c = 16 − 4 = 12 2
major : 8
c=2 3
minor : 4
sketch of graph on next slide

34. HW #4
“Of those to whom much is given, much is required.”
John F. Kennedy