2. Concepts and Objectives
Ellipses
Identify the equation of an ellipse
Find the center, x-radius, and y-radius of an ellipse
Find the major and minor axes
Find the foci and focal length of an ellipse
Write the equation of an ellipse
Solve problems involving ellipses
3. Ellipses
An ellipse, geometrically speaking, is a set of points in a
plane such that for each point, the sum of its distances,
d1 + d2, from two fixed points F1 and F2, is constant.
What does this mean?
4. Parts of an Ellipse
Parts of an ellipse:
Center
5. Parts of an Ellipse
Parts of an ellipse:
Center
Vertices (ea. vertex)
6. Parts of an Ellipse
Parts of an ellipse:
Center
Vertices (ea. vertex)
Major axis
7. Parts of an Ellipse
Parts of an ellipse:
Center
Vertices (ea. vertex)
Major axis
Minor axis
8. Parts of an Ellipse
Parts of an ellipse:
Center
Vertices (ea. vertex)
Major axis
Minor axis
Foci (ea. focus)
9. Parts of an Ellipse
Other important parts:
The semi-major and semi-minor axes are half the
length of the major and minor axes.
The distance from the center to the ellipse in the
x-direction is called the x-radius. Likewise, the
distance in the y-direction is called the y-radius.
The distance between the foci is called the focal
length. The distance between the center and a focus
is called the focal radius.
10. Standard Form
The standard form of an ellipse centered at h, k is
where rx is the x-radius and ry is the y-radius.
To graph an ellipse from the standard form, plot the
center, mark the x- and y-radii, and sketch in the curve.
The ellipse will be wider in the direction that has the
larger radius.
2
2
1
x y
x h y k
r r
12. Standard Form
Example: Sketch the graph of
The center is at 4, –1
2 2
4 1
1
3 5
x y
13. Standard Form
Example: Sketch the graph of
The center is at 4, –1
The x-radius is 3
2 2
4 1
1
3 5
x y
14. Standard Form
Example: Sketch the graph of
The center is at 4, –1
The x-radius is 3
The y-radius is 5
2 2
4 1
1
3 5
x y
15. Standard Form
Example: Sketch the graph of
The center is at 4, –1
The x-radius is 3
The y-radius is 5
Sketch in the curves
2 2
4 1
1
3 5
x y
16. General Form of an Ellipse
Example: Sketch the graph of
2 2
4 9 16 90 205 0
x y x y
17. General Form of an Ellipse
Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
2 2
4 9 16 90 205 0
x y x y
2 2
4 16 9 90 205
x x y y
18. General Form of an Ellipse
Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
2 2
4 9 16 90 205 0
x y x y
2 2
4 16 9 90 205
x x y y
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
Remember, you have to
factor out the
coefficients of x2 and y2!
19. General Form of an Ellipse
Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
2 2
4 9 16 90 205 0
x y x y
2 2
4 16 9 90 205
x x y y
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
2 2
4 2 9 5 36
x y
20. General Form of an Ellipse
Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
2 2
4 9 16 90 205 0
x y x y
2 2
4 16 9 90 205
x x y y
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
2 2
4 2 9 5 36
36 36 36
x y
21. General Form of an Ellipse
Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
2 2
4 9 16 90 205 0
x y x y
2 2
4 16 9 90 205
x x y y
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
2 2
4 2 9 5 36
36 36 36
x y
2 2
2 5
1
3 2
x y
2 2
2 5
1
9 4
x y
22. General Form of an Ellipse
Example (cont.):
The center is at 2, –5
The x-radius is 3 (semi-major)
The y-radius is 2 (semi-minor)
2 2
2 5
1
3 2
x y
23. Ellipses
With ellipses, we use the following notation:
a is the length of the semi-major axis
b is the length of the semi-minor axis
c is the length of the focal radius.
Therefore,
The length of the major axis is 2a
The length of the minor axis is 2b
The sum of the distances from a point x, y on the
ellipse to the two foci is 2a (the major axis length)
24. Focal Radius
Not only does the dotted
line trace the outline of the
ellipse, but it is also the
length of the major axis. As
you can see from the
picture, half of that length
(a) is the hypotenuse of the
triangle formed by the semi-
minor axis and the focal
radius. This gives us the
formula:
2 2 2
c a b
b
a
c
(careful!)
25. Focal Length
Example: Write the equation of the ellipse having center
at the origin, foci at –5, 0 and 5, 0, and major axis of
length 18 units.
26. Focal Length
Example: Write the equation of the ellipse having center
at the origin, foci at –5, 0 and 5, 0, and major axis of
length 18 units.
Since the major axis is 18 units long, 2a = 18, so a = 9.
The distance between the center and a focus, c, is 5.
Therefore, we can find b using the formula:
2 2 2
c a b
2 2 2
5 9 b
2 2 2
9 5 56
b
56
b
27. Focal Length
Example (cont.):
The foci lie along the major axis, so we know that rx = a.
Putting it all together, we have:
or
2
2
1
9 56
x y
2 2
1
81 56
x y
28. Eccentricity
The eccentricity of an ellipse is a measure of its
“roundness”, and it is the ratio of the focal length to the
major axis.
This ratio is written as
c
e
a
29. Eccentricity
Example: The orbit of Jupiter is an ellipse with the sun
at one focus (mostly). The eccentricity of the ellipse is
0.0489, and the maximum distance of Jupiter from the
sun is 507.4 million miles. Find the closest distance that
Jupiter comes to the sun.
30. Eccentricity
Example: The orbit of Jupiter is an ellipse with the sun
at one focus (mostly). The eccentricity of the ellipse is
0.0489, and the maximum distance of Jupiter from the
sun is 507.4 million miles. Find the closest distance that
Jupiter comes to the sun.
Jupiter
Sun
a + c
a – c
0.0489
c
a
0.0489
c a
507.4
a c
0.0489 507.4
a a
507.4
483.74
1.0489
a
