Holt Algebra 2

10-3 Ellipses 10-3 Ellipses

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

10-3 Ellipses

Warm Up

If c2 = a2 – b2, find c ifc = ±121. a = 13, b = 5

2. a = 4, b = 3

Holt Algebra 2

10-3 Ellipses

Write the standard equation for an ellipse.

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

Objectives

Holt Algebra 2

10-3 Ellipses

ellipsefocus of an ellipsemajor axisvertices of an ellipseminor axisco-vertices of an ellipse

Vocabulary

Holt Algebra 2

10-3 Ellipses

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.

Holt Algebra 2

10-3 Ellipses

Find the constant sum for an ellipse with foci F1 (3, 0) and F2 (24, 0) and the point on the ellipse (9, 8).

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

d = PF1 + PF2 Definition of the constant sum of an ellipseDistance Formula

Substitute.

Simplify.

d = 27

The constant sum is 27.

Holt Algebra 2

10-3 Ellipses

Find the constant sum for an ellipse with foci F1 (0, –8) and F2 (0, 8) and the point on the ellipse (0, 10).

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

Check It Out! Example 1

Distance Formula

Substitute.

d = 20

The constant sum is 20.

Simplify.324 4d

Holt Algebra 2

10-3 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.

Holt Algebra 2

10-3 Ellipses

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

Holt Algebra 2

10-3 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.

Holt Algebra 2

10-3 Ellipses

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

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

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

Step 1 Choose the appropriate form of equation.

The vertex is on the x-axis. x2

a2+ = 1 y2

b2

Holt Algebra 2

10-3 Ellipses

Example 2A Continued

Step 2 Identify the values of a and b.

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

x2

36+ = 1 y2

16

a = 6

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

Step 3 Write the equation.

Substitute the values into the equation of an ellipse.

Holt Algebra 2

10-3 Ellipses

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

Step 1 Choose the appropriate form of equation.

The vertex is on the y-axis. y2

a2+ = 1 x2

b2

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

Step 2 Identify the values of b and c.

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

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

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

Holt Algebra 2

10-3 Ellipses

y2

34+ = 1 x2

25

Step 3 Use the relationship c2 = a2 – b2 to find a2.

Substitute 3 for c and 5 for b.

Step 4 Write the equation.

32 = a2 – 52

a2 = 34

Example 2B Continued

Substitute the values into the equation of an ellipse.

Holt Algebra 2

10-3 Ellipses

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

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

Step 1 Choose the appropriate form of equation.

The vertex is on the x-axis. x2

a2+ = 1 y2

b2

Check It Out! Example 2a

Holt Algebra 2

10-3 Ellipses

Step 2 Identify the values of a and b.

The vertex (9, 0) gives the value of a.

x2

81+ = 1 y2

25

a = 9

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

Step 3 Write the equation.

Substitute the values into the equation of an ellipse.

Check It Out! Example 2a Continued

Holt Algebra 2

10-3 Ellipses

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

Co-vertex at (4, 0); focus at (0, 3)

Step 1 Choose the appropriate form of equation.

The vertex is on the y-axis. y2

a2+ = 1 x2

b2

Check It Out! Example 2b

Holt Algebra 2

10-3 Ellipses

Step 2 Identify the values of b and c.

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

y2

25+ = 1 x2

16

b = 4

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

Step 3 Use the relationship c2 = a2 – b2 to find a2.

Substitute 3 for c and 4 for b.

Check It Out! Example 2b Continued

Step 4 Write the equation.

32 = a2 – 42

a2 = 25

Holt Algebra 2

10-3 Ellipses

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

Holt Algebra 2

10-3 Ellipses

Example 3: Graphing Ellipses

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.

Graph the ellipse

Holt Algebra 2

10-3 Ellipses

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 3 Continued

Graph the ellipse

Holt Algebra 2

10-3 Ellipses

Step 1 Rewrite the equation as

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.

Check It Out! Example 3a

Graph the ellipse

Holt Algebra 2

10-3 Ellipses

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).

Check It Out! Example 3a Continued

Graph the ellipse

Holt Algebra 2

10-3 Ellipses

Graph the ellipse.

Step 1 Rewrite the equation as

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

h = 2 and k = 4, so the center is (2, 4).

a = 5 and b = 3; Because 5 > 3, the major axis is horizontal.

Check It Out! Example 3b

Holt Algebra 2

10-3 Ellipses

Step 3 The vertices are (2 ± 5, 4) or (7, 4) and (–3, 4), and the co-vertices are (2, 4 ± 3), or (2, 7) and (2, 1).

Check It Out! Example 3b Continued

(2, –1)

(7, 4)

Graph the ellipse.

Holt Algebra 2

10-3 Ellipses

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.

Example 4: Engineering Application

x2

50+ = 1 y2

20

Holt Algebra 2

10-3 Ellipses

Example 4 Continued

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

Holt Algebra 2

10-3 Ellipses

Step 2 Find the dimensions of the new park.

The width of the park is .

The length of the park is .

Example 4 Continued

Holt Algebra 2

10-3 Ellipses

B. Write an equation for the design of the new park.

For the new park, .

The equation in standard form for the new park

will be .

Step 1 Use the dimensions of the new park to find the values of a and b.

Step 2 Write the equation.

Example 4 Continued

Holt Algebra 2

10-3 Ellipses

Check It Out! Example 4

Engineers have designed a tunnel with

the equation measured in

feet. A design for a larger tunnel needs to

be twice as wide and 3 times as tall.

x2

64+ = 1, y2

36

Holt Algebra 2

10-3 Ellipses

a2 = 64, so a = 8 and the width of the tunnel is 2a = 16 ft.

b2 = 36, so b = 6 and the height of the tunnel is 6 ft.

Check It Out! Example 4 Continued

a. Find the dimensions for the larger tunnel.

Step 1 Find the dimensions of the original tunnel. Because 64 > 36, the major axis of the tunnel is horizontal.

Step 2 Find the dimensions of the larger tunnel.

The height of the larger tunnel is 3(6) = 18 ft.

The width of the larger tunnel is 2(16) = 32 ft.

Holt Algebra 2

10-3 Ellipses

b. Write an equation for the larger tunnel.

For the larger tunnel, a = 16 and b = 18.

Check It Out! Example 4 Continued

The equation in standard form for the larger tunnel

is x2

162+ = 1 or + = 1. y2

182 x2

256 y2

324

Step 1 Use the dimensions of the larger tunnel to find the values of a and b.

Step 2 Write the equation.

Holt Algebra 2

10-3 Ellipses

Lesson Quiz: Part I

16

1. Find the constant sum for an ellipse with foci F1 (2, 0), F2(–6, 0) and the point on the ellipse (2, 6).

2. Write an equation in standard form for each ellipse with the center at the origin.

A. Vertex at (0, 5); co-vertex at (1, 0).

B. Vertex at (5, 0); focus at (–2, 0).

Holt Algebra 2

10-3 Ellipses

Lesson Quiz: Part II

3. Graph the ellipse .