An ellipse is defined algebraically as the set of all points where the sum of the distances to two fixed points (the foci) is a constant. Geometrically, an ellipse can be constructed by stretching a circle: using a piece of string fixed at both ends (the foci) and tracing the path of a pencil as it is moved around so that the total length of string remains constant. The standard equation of an ellipse is (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center and a and b are the lengths of the semi-major and semi-minor axes. To graph an ellipse, one plots

1. Conics

2. Ellipses
An ellipse is the locus of a variable point on a
plane so that the sum of its distance from two
fixed points is a constant.
P’(x,y)
P’’(x,y)

3. Let PF1+PF2 = 2a where a > 0
aycxycx 2)()( 2222
=++++−
2222
)(2)( ycxaycx ++−=+−
222222
)()(44)( ycxycxaaycx +++++−=+−
222
44)(4 acxycxa +=++
42222222
2)2( acxaxcycxcxa ++=+++
42222222222
22 acxaxcyacaxcaxa ++=+++

4. 22422222
)( caayaxca −=+−
)()( 22222222
caayaxca −=+−
222
cabLet −=
222222
bayaxb =+
12
2
2
2
=+
b
y
a
x standard equation of
an ellipse

5. vertex
major axis = 2a
minor axis = 2b
lactus rectum
length of semi-major axis = a
length of the semi-minor axis = b
length of lactus rectum =
a
b2
2

7. Other form of Ellipse
12
2
2
2
=+
a
y
b
x

8. Ellipse
• Review: The geometric definition relies on
a cone and a plane intersecting it
• Algebraic definition: a set of points in the
plane such that the sum of the distances
from two fixed points, called foci, remains
constant.

9. Hands-on Activity
At your table is paper, corkboard, string, and
tacks.
Follow the directions on your handout to
complete the activity.

10. Ellipse Definitions
Algebraic Definition of an Ellipse: a set of points in the
plane such that the sum of the distances from two fixed
points, called foci, remains constant.
 What remains constant in your sketch?
 The points where you placed the tacks are known as the
foci. Draw a line through f1 and f2 to the edges of the
ellipse. This is known as the major axis. Locate the
midpoint between f1 and f2. Is this the center of the
ellipse? Will that always be the case?
 What inference can you draw from the data?
 Does the data support the definition? Explain.

11. Facts: Ellipse Equation
Both variables are squared.
Equation:
What makes the ellipse different from the
circle?
What makes the ellipse different from the
parabola?
2 2
2 2
( - ) ( - )
1
x h y k
a b
+ =

12. where the center is at (h,k) and |2a | is the length of the
horizontal axis and |2b| is the of the length of the
vertical axis.
Procedure to graph:
1. Put in standard form (above): x squared term
+ y squared term = 1
2. Plot the center (h,k)
3. Plot the endpoints of the horizontal axis by
moving “a” units left and right from the
center.
2 2
2 2
( - ) ( - )
1
x h y k
a b
+ =

13. where the center is at (h,k) and |2a | is the length of the
horizontal axis and |2b| is the of the length of the
vertical axis.
To graph:
4. Plot the endpoints of the vertical axis by moving
“b” units up and down from the center.
Note: Steps 3 and 4 locate the endpoints of the
major and minor axes.
5. Connect endpoint of axes with smooth curve.
2 2
2 2
( - ) ( - )
1
x h y k
a b
+ =

14. where the center is at (h,k) and |2a | is the length of the
horizontal axis and |2b| is the of the length of the
vertical axis.
To graph:
6. Use the following formula to help locate the
foci: c2
= a2
- b2
if a>b or c2
= b2
– a2
if b>a
**Challenge question: Why are we using this
formula to locate the foci? Draw a diagram and
justify your answer.**
2 2
2 2
( - ) ( - )
1
x h y k
a b
+ =

15. where the center is at (h,k) and |2a | is the length of the
horizontal axis and |2b| is the of the length of the
vertical axis.
To graph:
6. (continued) Move “c” units left and right form
the center if the major axis is horizontal
OR Move “c” units up and down form the center if
the major axis is vertical
Label the points f1 and f2 for the two foci.
Note: It is not necessary to plot the foci to graph the ellipse, but it is common
practice to locate them.
2 2
2 2
( - ) ( - )
1
x h y k
a b
+ =

16. where the center is at (h,k) and |2a | is the length of the
horizontal axis and |2b| is the of the length of the
vertical axis.
To graph:
7. Identify the length of the major and minor
axes.
2 2
2 2
( - ) ( - )
1
x h y k
a b
+ =

17. Exp. 1: Graph
To graph:
1. Put in standard form (set = 1)
Done
2. Plot the center (h,k)
(-2,3)
3. Plot the endpoints of the horizontal axis by
moving “a” units left and right from the center.
Endpoints at (-7,3) and (3,3)
2 2
( 2) ( -3)
1
25 16
x y+
+ =

18. Exp. 1: Graph
4. Plot the endpoints of the vertical axis by
moving “b” units up and down from the
center.
Endpoints at (-2,7) and (-2,-1)
5. Connect endpoint of axes with smooth curve
2 2
( 2) ( -3)
1
25 16
x y+
+ =

19. Major axis
Center
Minor axis
2 2
( 2) ( -3)
1
25 16
x y+
+ =

20. Exp. 1: Graph
6. Which way is the major axis in this problem (horizontal or
vertical)?
Horizontal because 25>16 and 25 is under the “x”
Use the following formula to help locate the foci: c2
= a2
- b2
if
a>b or c2
= b2
– a2
if b>a
c2
= a2
- b2
c2
= 25– 16
c2
= 9
c= ±3
Move 3 units left and right from the center to locate the foci.
Where are the foci?
(-5,3) and (1,3)
2 2
( 2) ( -3)
1
25 16
x y+
+ =

21. Foci
f1 f2
Length of Major Axis is 10.
Length of Minor Axis is 8.
2 2
( 2) ( -3)
1
25 16
x y+
+ =

22. Exp. 2: Graph 16x2
+ 9y2
= 144
To graph:
1. Put in standard form.
2. Plot the center
(0,0)
3. Plot the endpoints of the horizontal axis.
Endpoints at (-3,0) and (3,0)
2 2
1
9 16
x y
+ =

23. Exp. 2: Graph 16x2
+ 9y2
= 144
4. Plot the endpoints of the vertical axis.
Endpoints at (0,4) and (0,-4)
5. Connect endpoint of axes with smooth curve
6. Which way is the major axis in this problem?
Vertical because 16>9 and 16 is under the “y”
Locate the foci:
c2
= b 2
- a2
c2
= 16 - 9
c2
= 7
c= ±√7
Where are the foci?
(0, √7) and (0,-√7)
2 2
1
9 16
x y
+ =

24. 2 2
1
9 16
x y
+ =
Length of Major Axis is 8.
Length of Minor Axis is 6.

25. Exp. 3: Graph 4x2
+ 9y2
+ 16x – 54y +61 = 0.
1. Put in standard form.
(Hint: Complete the square.)
4x2
+ 16x + 9y2
– 54y = -61
4(x2
+ 4x ) + 9(y2
– 6y ) = -61
+4 +9 +16 + 81
4(x + 2)2
+ 9(y – 3)2
= 36
2. Plot the center
(-2,3)
3. Plot the endpoints of the horizontal axis.
Endpoints at (-5,3) and (1,3)
2 2
( 2) ( 3)
1
9 4
x y+ −
+ =

26. Exp. 3: Graph 4x2
+ 9y2
+ 16x – 54y +61 = 0.
4. Plot the endpoints of the vertical axis.
Endpoints at (-2,5) and (-2,1)
5. Connect endpoint of axes with smooth curve
6. Which way is the major axis in this problem?
Horizontal
Locate the foci:
c2
= a2
- b2
c2
= 9 - 4
c2
= 5
c= ±√5
Where are the foci?
(-2 ±√5, 3)
2 2
( 2) ( 3)
1
9 4
x y+ −
+ =

27. Length of Major Axis is 6.
Length of Minor Axis is 4.
2 2
( 2) ( 3)
1
9 4
x y+ −
+ =

28. Challenge Question
Given the following information, write the
equation of the ellipse. Sketch and find the
foci.
Center is (4,-3), the major axis is vertical
and has a length of 12, and the minor axis
has a length of 8.

29. Review
1) How can you tell if the graph of an
equation will be a line, parabola, circle, or
an ellipse?
2) What’s the standard form of an ellipse?
3) What are the steps for graphing an
ellipse?
4) What’s the standard form of a parabola?
5) What’s the standard form of a circle?
6) How are the various equations similar and
different?