The document provides an overview of ellipses, detailing their geometric properties, standard and general forms, and the equations used to represent them. It includes definitions of key components such as center, vertices, axes, foci, and eccentricity, along with examples illustrating how to graph and calculate properties of ellipses. Additionally, it mentions related algebraic problems for practice.

1. 10.2 Ellipses
Chapter 10 Analytic Geometry

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
⚫ Vertices (ea. vertex)
⚫ Major axis
⚫ Minor axis
⚫ Foci (ea. focus)
• •

5. 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.

6. 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.
  − −
+ =    
   
22
1
x y
x h y k
r r

7. Standard Form
⚫ Example: Sketch the graph of
− +   
+ =   
   
2 2
4 1
1
3 5
x y

8. 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

9. General Form of an Ellipse
⚫ Example: Sketch the graph of
+ − + + =2 2
4 9 16 90 205 0x y x y

10. ⚫ Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
General Form of an Ellipse
+ − + + =2 2
4 9 16 90 205 0x y x y
( ) ( )− + + = −2 2
4 16 9 90 205x x y y

11. ⚫ Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
General Form of an Ellipse
+ − + + =2 2
4 9 16 90 205 0x y x y
( ) ( )− + + = −2 2
4 16 9 90 205x x y y
( ) ( ) ( ) ( )− + + = −+ ++ +2 2222 2
4 4 10 2052 4 29 5 9 5x x y y
Remember, you
have to factor out
the coefficient of x2!

12. ⚫ Example: Sketch the graph of
To sketch the graph, we have to rewrite the equation:
General Form of an Ellipse
+ − + + =2 2
4 9 16 90 205 0x y x y
( ) ( )− + + = −2 2
4 16 9 90 205x x y y
( ) ( ) ( ) ( )− + + = −+ ++ +2 2222 2
4 4 10 2052 4 29 5 9 5x x y y
( ) ( )− + + =
2 2
4 2 9 5 36x y
Remember, you
have to factor out
the coefficient of x2!

13. ( ) ( )− +
+ =
2 2
4 2 9 5 36
36 36 36
x y
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 0x y x y
( ) ( )− + + = −2 2
4 16 9 90 205x x y y
( ) ( ) ( ) ( )− + + = −+ ++ +2 2222 2
4 4 10 2052 4 29 5 9 5x x y y
Remember, you
have to factor out
the coefficient of x2!

14. 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 0x y x y
( ) ( )− + + = −2 2
4 16 9 90 205x x y y
( ) ( ) ( ) ( )− + + = −+ ++ +2 2222 2
4 4 10 2052 4 29 5 9 5x 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
Remember, you
have to factor out
the coefficient of x2!


15. 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

16. 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

17. 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!)

18. 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.

19. 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 56b
= 56b

20. Focal Length
⚫ Example (cont.):
The foci lie along the major axis, so we know that rx = a.
Putting it all together, we have:
or
  
+ =   
   
22
1
9 56
x y
+ =
2 2
1
81 56
x y

21. 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

22. 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.

23. 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
=0.0489c a
+ =507.4a c
+ =0.0489 507.4a a
= 
507.4
483.74
1.0489
a

24. Eccentricity
⚫ Example (cont.):
•
•
Jupiter
Sun
a + c
a – c
( )=0.0489 483.74c
=23.65c
− = −483.74 23.65a c
= 460.1 million miles

25. Classwork
⚫ College Algebra
⚫ Page 968: 4-16 (even), page 957: 20-38 (even),
page 200: 38-42 (even)