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Holt Algebra 2
10-3 Ellipses 10-3 Ellipses
Holt Algebra 2
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
10-3 Ellipses
Warm Up
If c2 = a
2 – b
2, find c if
c = ±12 1. 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
ellipse
focus of an ellipse
major axis
vertices of an ellipse
minor axis
co-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 ellipse
Distance
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 c
2 = a
2 – b
2. 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 y
2
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 y
2
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 x
2
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 x
2
25
Step 3 Use the relationship c2 = a
2 – b
2 to find a
2.
Substitute 3 for c and 5 for b.
Step 4 Write the equation.
32 = a
2 – 5
2
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 y
2
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 y
2
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 x
2
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 x
2
16
b = 4
c = 3 The focus (0, 3) gives the value of c.
Step 3 Use the relationship c2 = a
2 – b
2 to find a
2.
Substitute 3 for c and 4 for b.
Check It Out! Example 2b Continued
Step 4 Write the equation.
32 = a
2 – 4
2
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. y
2
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 .
