11.3 Ellipses

Objective: By the end of the lesson, you should be able to write an equation of an ellipse and

sketch its graph.

Table of ContentsDefinition of an Ellipse

Definition of Foci

Definition of Vertices

Definition of Major Axis

Definition of Minor Axis

Definition of Co-vertices

Standard Equation of an Ellipse

Vertical Major Axis

Calculating the Coordinates of the Foci

Eccentricity

Vocabulary you need to know

• Ellipse – The set of all points (x,y) such that the sum of the distances between (x,y) and two distinct fixed points (foci) are constant.

Cabri

Vocabulary you need to know

• Foci (plural of focus) - The two distinct points in an ellipse.

Focus Focus

Vocabulary you need to know

• Vertices – The two points at the intersection of the line through the foci and the ellipse.

Cabri

Vocabulary you need to know

• Vertices – The two points at the intersection of the line through the foci and the ellipse.

Vertex Vertex

Vocabulary you need to know

• Major Axis – The line segment joining the vertices

Major Axis

Vocabulary you need to know

• Minor axis – The line segment perpendicular to the major axis at the midpoint.

Minor Axis

Vocabulary you need to know

• Co-vertices – The endpoints of the minor axis

Co-vertex

Co-vertex

In a general ellipse with the center at the origin…

• a is the distance from the center to the vertex• b is the distance from the center to the co-vertex

a

b

Cabri

In a general ellipse…

So the coordinate of the right vertex is (a,0).

The coordinate of the left vertex is (-a,0).

(-a,0) (a,0)aa

2a Cabri

In a general ellipse

The coordinate of the top co-vertex is (0,b)

The coordinate of the bottom vertex is (0,-b)

(0,b)

(0,-b)

b

b

2b

Standard equation of an ellipse with the center at the origin

• The standard form of the equation of a ellipse with center at (0,0) and major and minor axes of length 2a and 2b, where a>b, is as follows.

12

2

2

2

b

y

a

x

12

2

2

2

a

y

b

x

Horizontal major axis

Vertical major axis

Cabri

Standard equation of an ellipse with the center at the origin

12

2

2

2

b

y

a

x

12

2

2

2

a

y

b

x

Horizontal major axis

Vertical major axis

Notice the switch!!

What does an ellipse with a vertical major axis look like?

Major axis

Minor axis

Vertex

Vertex

Co-vertex Co-vertex

Focus

Focus

How do we calculate the coordinates of the foci?

• The foci of the ellipse lie on the major axis, c units from the center, where c2=a2-b2

Cabri

How do we calculate the coordinates of the foci?

• The foci of the ellipse lie on the major axis, c units from the center, where c2=a2-b2

c c

(c,0)(-c,0)

What is Eccentricity?

Eccentricity tells us how flat (or round) the ellipse is.

Cabri

a

ce

What is Eccentricity?

Eccentricity tells us how flat (or round) the ellipse is.

a

ce

As e approaches 0, the ellipse becomes a circle.

As e approaches 1, the ellipse flattens to a line.

Example 1 Write an equation of an ellipse whose vertices

are (-5,0) and (5,0) and whose co-vertices are (0,-3) and (0,3). Find the foci of the ellipse.