Transcript Conic Sections Ellipses - University of South Florida

Colleen Beaudoin
January, 2009


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.
A
B
d1
d1
x
From each
point in the
plane, the sum
of the distances
to the foci is a
constant.
d2
f1
d2
Example:
f2
Point A: d1+d2 = c
foci
y
Point B: d1+d2 = c
Center
Major axis
x
f1
Minor axis
f2
foci
y
At your table is paper, corkboard, string,
and tacks.
Follow the directions on your handout to
complete the activity.
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.




Both variables are squared.
Equation:
What makes the ellipse different from the
circle?
What makes the ellipse different from the
parabola?
2
2
( x - h) ( y - k )

1
2
2
a
b
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
( x - h) ( y - k )

1
2
2
a
b
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
( x - h) ( y - k )

1
2
2
a
b
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
( x - h) ( y - k )

1
2
2
a
b
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
( x - h) ( y - k )

1
2
2
a
b
To graph:
7. Identify the length of the major and minor
axes.
( x  2) ( y - 3)

1
25
16
2
2
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)
( x  2) ( y - 3)

1
25
16
2
2
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
( x  2) ( y - 3)

1
25
16
2
2
Major axis
Center
Minor axis
( x  2) ( y - 3)

1
25
16
2
2
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)
( x  2) ( y - 3)

1
25
16
2
2
Foci
f1
f2
Length of Major Axis is 10.
Length of Minor Axis is 8.
To graph:
1. Put in standard form.
2
2
x
y

1
9 16
2. Plot the center
(0,0)
3. Plot the endpoints of the horizontal axis.
Endpoints at (-3,0) and (3,0)
x2 y 2

1
9 16
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
x
y

1
9 16
Length of Major Axis is 8.
Length of Minor Axis is 6.
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
( x  2)2 ( y  3)2

1
9
4
2. Plot the center
(-2,3)
3. Plot the endpoints of the horizontal axis.
Endpoints at (-5,3) and (1,3)
( x  2)2 ( y  3)2

1
9
4
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)
( x  2)2 ( y  3)2

1
9
4
Length of Major Axis is 6.
Length of Minor Axis is 4.
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.
1)
2)
3)
4)
5)
6)
How can you tell if the graph of an equation
will be a line, parabola, circle, or an ellipse?
What’s the standard form of an ellipse?
What are the steps for graphing an ellipse?
What’s the standard form of a parabola?
What’s the standard form of a circle?
How are the various equations similar and
different?