Chapter 12 12-3 Ellipses

Objectives

Write the standard equation for an ellipse.

Graph an ellipse, and identify its center, vertices, co-vertices, and foci.

Ellipse

If you pulled the center of a circle apart into two points, it would stretch the circle into an ellipse.

An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F1 and F2, called the foci (singular: focus), is the constant sum d = PF1 + PF2. This distance d can be represented by the length of a piece of string connecting two pushpins located at the foci.

You can use the distance formula to find the constant sum of an ellipse.

Definition of constant sum

It is the sum of the distances from the focus to a point on the ellipse

Example 1: Using the Distance Formula to Find the Constant Sum of an Ellipse

Find the constant sum for an ellipse with foci F1 (3, 0) and F2

(24, 0) and the point on the ellipse (9, 8).

Solution:

d = PF1 + PF2 Definition of the constant sum of an ellipse

Distance Formula

d = 27

The constant sum is 27

Example

Find the constant sum for an ellipse with foci F1 (0, –8) and F2

(0, 8) and the point on the ellipse (0, 10).

Solution:

The constant sum is 20.

ELLIPSES

Instead of a single radius, an ellipse has two axes. The longer the axis of an ellipse is the major axis and passes through both foci. The endpoints of the major axis are the vertices of the ellipse. The shorter axis of an ellipse is the minor axis. The endpoints of the minor axis are the co-vertices of the ellipse. The major axis and minor axis are perpendicular and intersect at the center of the ellipse.

ELLIPSES

The standard form of an ellipse centered at (0, 0) depends on whether the major axis is horizontal or vertical.

ELLIPSES

The values a, b, and c are related by the equation c2 = a2 – b2. Also note that the length of the major axis is 2a, the length of the minor axis is 2b, and a > b.

Example 2A: Using Standard Form to Write an Equation for an Ellipse

Write an equation in standard form for each ellipse with center (0, 0).

Vertex at (6, 0); co-vertex at (0, 4)

SOLUTION:

Step 1 Choose the appropriate form of equation.

x2

a2+ = 1 y2

b2The vertex is on the x-axis.

SOLUTION

Step 2 Identify the values of a and b.

a = 6 The vertex (6, 0) gives the value of a.

b = 4 The co-vertex (0, 4) gives the value of b.

Step 3 Write the equation.

x2

36+ = 1 y2

16

EXAMPLE

Write an equation in standard form for each ellipse with center (0, 0).

Co-vertex at (5, 0); focus at (0, 3)

Solution:

Step 1 Choose the appropriate form of equation. y2

a2+ = 1 x2

b2The vertex is on the y-axis.

Step 2 Identify the values of b and c.

b = 5 The co-vertex (5, 0) gives the value of b.

c = 3 The focus (0, 3) gives the value of c.

solution

Step 3 Use the relationship c2 = a2 – b2 to find a2. 32 = a2 – 52 Substitute 3 for c and 5 for b.

a2 = 34

Step 4 Write the equation

y2

34+ = 1 x2

25

example

Write an equation in standard form for each ellipse with center (0, 0).

Vertex at (9, 0); co-vertex at (0, 5)

Ellipses with a center not in the origin

Ellipses may also be translated so that the center is not the origin.

Example 3: Graphing Ellipses

Graph the ellipse

Solution:

Step 1 Rewrite the equation as

Step 2 Identify the values of h, k, a, and b

h = –4 and k = 3, so the center is (–4, 3).

a = 7 and b = 4; Because 7 > 4, the major axis is horizontal.

solution

Step 3 The vertices are (–4 ± 7, 3) or (3, 3) and (–11, 3), and the co-vertices are (–4, 3 ± 4), or (–4, 7) and (–4, –1).

Example

Solution:

Step 1 Rewrite the equation as

Graph the ellipse

solution

Step 2 Identify the values of h, k, a, and b.

h = 0 and k = 0, so the center is (0, 0).

a = 8 and b = 5; Because 8 > 5, the major axis is horizontal.

Step 3 The vertices are (±8, 0) or (8, 0) and (–8, 0), and the co-vertices are (0, ±5,), or (0, 5) and (0, –5).

Example 4: Engineering Application

A city park in the form of an ellipse with equation , measured in meters, is being renovated. The new park will have a length and width double that of the original park.

Solution:

Find the dimensions of the new park

Step 1 Find the dimensions of the original park. Because 50 > 20, the major axis of the park is horizontal.

a2 = 50, so and the length of the park is .

b2 = 20, so and the width of the park is

solution

Step 2 Find the dimensions of the new park.

The length of the park is .

The width of the park is .

Write an equation for the design of the new park.

The equation in standard form for the new park

will be .

Student guided practice

Do even problems from 2-10 in your book page 834

Homework

Do problems from 13-21 in your book page 834

closure

Today we learned about ellipses

Next class we are going to learn about hyperbolas