Objectives:
1. Define an ellipse.
2. Write the equation of an ellipse.
3. Identify important characteristics of an ellipse.
4. Graph ellipses.
Ellipse
 The set of all points whose distances from two fixed
points add to the same constant. The two fixed points
are called the foci.
Standard Equation of an Ellipse
Centered at the Origin
x
2
a
2

y
2
b
2
1
Depending on which denominator is larger the ellipse could be
elongated along the x-axis or the y-axis. If a and b are equal, the
ellipse becomes a circle. For instance, given a circle of radius 2
centered at the origin, the equation is:
x2  y2  4
When everything is divided by 4, the equation becomes:
2
2
x
y

1
4
4
Characteristics of an Ellipse
Important
Facts:






The foci are within the
ellipse
The major axis always
contains the foci and is
determined by which
denominator is larger
The distance between the
foci is 2c
The center is the
midpoint of the foci and
the midpoint of the
vertices.
The distance between the
vertices are 2a and 2b.
c2 = a2 – b2
Example #1
 Write the following ellipse in standard form. Then graph and
label the foci, the vertices, & the major and minor axes.
2
2
9 x  25 y  225
9 x 2 25 y 2 225


225 225 225
x2 y2

1
25 9
x2
5
2

y2
3
2
1
c  a2  b2
 25  9
 16
4
Example #1
 Write the following ellipse in standard form. Then graph and
label the foci, the vertices, & the major and minor axes.
2
2
9 x  25 y  225
y
6
5
4
3
Vertices: (±5,0)
Covertices: (0,±3)
Foci: (±4,0)
Major Axis: x-axis
Minor Axis: y-axis
2
1
–6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
1
2
3
4
5
6
x
Example #2
 Solve the following equation for y and graph it using a graphing
calculator.
2
2
25x  4 y  100
4 y 2  100  25 x 2
2
100

25
x
y2 
4
100  25 x 2
y
4
Example #3A
A. Find the equation of an ellipse with vertices at (±8, 0)
and foci at  3 5 ,0


Then sketch its graph using the intercepts.
y
a  8, c  3 5
3 5  8 b
2
3 5 
2

9
8
2
 8 b 
2
45  64  b 2
 19  b
2
b  19  4.4
2
2
x
2
8
2
2

y
7
2
 19 
2
2
x
y

1
64 19
1
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
1
2
3
4
5
6
7
8
9
x
Example #3B
Find the equation of an ellipse with foci on the y-axis. The major
axis has a length of 10 and the minor axis has a length of 9. Then
sketch the graph.
x2
y2
 2 1
2
4.5 5
The length of the axes tells us the
B.
distance between the vertices.
Since the major axis is 10 units long
and on the y-axis, the vertices are
at (0, ±5).
For the minor axis, the covertices
are on the x-axis at (±4.5, 0).
This implies a = 5 and b = 4.5
x2
9
 
2
2

y2
5
2
1
x2 y2

1
81 25
4
4x2 y 2

1
81 25
Example #3B
B.
Find the equation of an ellipse with foci on the y-axis. The major
axis has a length of 10 and the minor axis has a length of 9. Then
sketch the graph.
y
9
8
x
2
4.5
2

y
2
5
2
7
1
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
1
2
3
4
5
6
7
8
9
x
Example #4
 A furniture maker has a rectangular block of wood that measures 37 ½
inches by 25 inches. She wants to cut it to make the largest elliptical
table top possible. Find an equation of an ellipse she can use, placing
the center at the origin and the major axis on the x-axis. Locate the foci
and sketch the graph.
37 ½
25
The length of the major
and minor axes are
basically given from the
dimensions of the
rectangular piece of wood.
This problems works very
similarly to the last
example.
Example #4
 Find an equation of an ellipse she can use, placing the center at the
origin and the major axis on the x-axis. Locate the foci and sketch the
graph.
37 ½
y
x
25
2
18.75 2

y
2
12.5 2
20
1
16
12
8
a = 37.5 ÷ 2 = 18.75
4
–20 –16 –12
Foci: (±14,0)
b = 25 ÷ 2 = 12.5
c  18.75  12.5
2
c  195.3125
c  14.0
–8
–4
4
–4
–8
–12
2
–16
–20
8
12
16
20
x
Example #5
 Sirius Radio has 3 satellites that travel in the same
elliptical orbit around the earth. The length of the
major axis of the orbit is 52,000 miles, the length of
the minor axis of the orbit is 50,000 miles, and Earth is
at one focus. Find the minimum and maximum
distances from any one of the three satellites to Earth.
Note:
The maximum distance is at a + c and
the minimum distance is at a – c.
Example #5
 Major axis 52,000 miles; minor axis is 50,000 miles.
Find the minimum and maximum distances from any
one of the three satellites to Earth.
c  26,000 2  25,000 2
 676,000,000  625,000,000
 51,000,000
 7141
a = 52,000 ÷ 2 = 26,000
b = 50,000 ÷ 2 = 25,000
Max : 26,000  7141  33,141 miles
Min : 26,000  7141  18,859 miles