10-3 Ellipses
Objectives and Vocabulary
Write the standard equation for an ellipse.
Graph an ellipse, and identify its center,
vertices, co-vertices, and foci.
ellipse
focus of an ellipse
major axis
minor axis
vertices and co-vertices of an ellipse
Holt Algebra 2
10-3 Ellipses
Notes
1. Graph the ellipse
.
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).
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)
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.
Holt Algebra 2
10-3 Ellipses
Example 1 : Graphing Ellipses
Graph the ellipse
The vertices are (±8, 0) or
(8, 0) and (–8, 0), and the
co-vertices are (0, ±5,), or
(0, 5) and (0, –5).
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
Example 2: Graphing Ellipses
Graph the ellipse
Step 3 The vertices are
(–4 ± 7, 3) or (3, 3) and
(–11, 3), and the covertices are (–4, 3 ± 4),
or (–4, 7) and (–4, –1).
Holt Algebra 2
10-3 Ellipses
Example 3A: 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)
Step 1
Step 2
Holt Algebra 2
x2
a2
+
y2
2 = 1
b
2
x2 + y = 1
16
36
Identify major axis The vertex is on the x-axis.
Substitute the values into the
equation of an ellipse.
10-3 Ellipses
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).
Co-vertex at (5, 0); focus at (0, 3)
Step 1 Choose the appropriate form of equation.
y2
a2
+
x2
2 = 1 The vertex is on the y-axis.
5
Step 2 Use the foci to complete equation
2
y2 + x = 1
25
34
Holt Algebra 2
10-3 Ellipses
Notes
1. Graph the ellipse
.
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).
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)
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.
Holt Algebra 2
10-3 Ellipses
Notes
1. Graph the ellipse
.
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).
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 #4: Engineering Application
4. A city park in the form of an ellipse with
2
y2
x
equation
+
= 1 , measured in
25
49
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.
Holt Algebra 2
10-3 Ellipses
Step 1: Find the dimensions of the existing park.
2
x2 + y = 1
25
49
Length of major axis = 14
Length of minor axis = 10
Step 2: Find the dimensions of the new park.
Length of major axis = 28
Length of minor axis = 20
Step 3: Write the equation for the new park.
x2 +
198
Holt Algebra 2
y2
100
=1
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
Ellipses: Extra Info
The following power-point slides contain extra
examples and information.
Reminder: Lesson Objectives
Write the standard equation for an ellipse.
Graph an ellipse, and identify its center, vertices, covertices, and foci.
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 covertices 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
Check It Out! Example 2a
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.
x2
a2
Holt Algebra 2
+
y2
2 = 1
b
The vertex is on the x-axis.
10-3 Ellipses
Check It Out! Example 2a Continued
Step 2 Identify the values of a and b.
a=9
The vertex (9, 0) gives the value of a.
b=5
The co-vertex (0, 5) gives the value of b.
Step 3 Write the equation.
2
x2 + y = 1
25
81
Holt Algebra 2
Substitute the values into the equation of
an ellipse.
10-3 Ellipses
Check It Out! Example 2b
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.
y2
a2
Holt Algebra 2
+
x2
2 = 1
b
The vertex is on the y-axis.
10-3 Ellipses
Check It Out! Example 2b Continued
Step 2 Identify the values of b and c.
b=4
The co-vertex (4, 0) gives the value of b.
c=3
The focus (0, 3) gives the value of c.
Step 3 Use the relationship c2 = a2 – b2 to find a2.
32 = a2 – 42 Substitute 3 for c and 4 for b.
a2 = 25
Step 4 Write the equation.
2
y2 + x = 1
16
25
Holt Algebra 2
10-3 Ellipses
Check It Out! Example 3b
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.
Holt Algebra 2
10-3 Ellipses
Check It Out! Example 3b Continued
Graph the ellipse.
Step 3 The vertices are
(2 ± 5, 4) or (7, 4) and
(–3, 4), and the covertices are (2, 4 ± 3),
or (2, 7) and (2, 1).
Holt Algebra 2
(7, 4)
(2, –1)