holt algebra 2 10-3 ellipses write the standard equation for an ellipse. graph an ellipse, and...
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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 and Vocabulary
ellipse focus of an ellipsemajor axis minor axisvertices and co-vertices of an ellipse
Holt Algebra 2
10-3 Ellipses Notes
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).
4. A city park in the form of an ellipse is being renovated. The new park will have a length and width double that of the original park. Write the equation for the new park.
1. Graph the ellipse .
3. Write an equation of a shifted ellipse with the vertices (2, 5) and (2, -3) and co-Vertices of (4, 1) and (0, 1)
Holt Algebra 2
10-3 Ellipses
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 1 : Graphing Ellipses
Graph 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
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
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 2: Graphing Ellipses
Graph the ellipse
Holt Algebra 2
10-3 Ellipses
Write an equation in standard form for each ellipse with center (0, 0).
Example 3A: Using Standard Form to Write an Equation for an Ellipse
Vertex at (6, 0); co-vertex at (0, 4)
Step 1 Identify major axis -
The vertex is on the x-axis.
x2
a2+ = 1 y2
b2
x2
36+ = 1 y2
16Substitute the values into the equation of an ellipse.
Step 2
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
52
Example 3B: Using Standard Form to Write an Equation for an Ellipse
Write an equation in standard form for each ellipse with center (0, 0).
y2
34+ = 1 x2
25
Step 2 Use the foci to complete equation
Holt Algebra 2
10-3 Ellipses Notes
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).
4. A city park in the form of an ellipse is being renovated. The new park will have a length and width double that of the original park. Write the equation for the new park.
1. Graph the ellipse .
3. Write an equation of a shifted ellipse with the vertices (2, 5) and (2, -3) and co-Vertices of (4, 1) and (0, 1)
Holt Algebra 2
10-3 Ellipses
Notes
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).
1. Graph the ellipse .
3. Write an equation of a shifted ellipse with the vertices (2, 5) and (2, -3) and co-Vertices of (4, 1) and (0, 1)
Holt Algebra 2
10-3 Ellipses
4. 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. Write the equation for the new park.
Notes #4: Engineering Application
x2
49+ = 1 y2
25
Holt Algebra 2
10-3 Ellipses
Step 1: Find the dimensions of the existing park.
Step 2: Find the dimensions of the new park.
Step 3: Write the equation for the new park.
x2
49+ = 1 y2
25
Length of major axis = 14 Length of minor axis = 10
Length of major axis = 28Length of minor axis = 20
x2
198+ = 1 y2
100
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
Ellipses may also be translated so that the center is not the origin.
Holt Algebra 2
10-3 Ellipses
Reminder: Lesson ObjectivesWrite the standard equation for an ellipse.
Graph an ellipse, and identify its center, vertices, co-vertices, and foci.
Ellipses: Extra Info
The following power-point slides contain extra examples and information.
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
Notes #3: Solutions
3. Graph the 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
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.