Ellipses
| Site: | Clare-Gladwin RESD |
| Course: | Michigan Algebra II KHauck |
| Book: | Ellipses |
| Printed by: | Guest user |
| Date: | Tuesday, November 26, 2024, 12:28 AM |
Description
Ellipses
An ellipse is the set of points in a plane such that the sum of the distances from two fixed points in that plane stays constant. The two points are each called a focus point. The plural of focus is foci. The midpoint of the segment joining the foci is called the center of the ellipse. An ellipse has two axes of symmetry. The longer one is called the major axis, and the shorter one is called the minor axis. The two axes intersect at the center of the ellipse. When the length of the major axis is the same as the minor axis, the ellipse is a circle.
Horizontal
The equation of an ellipse that is centered at (0, 0) and has its major axis along the x-axis has the following standard form:
The length of the major axis is 2|a|, and the length of the minor axis is 2|b|. The endpoints of the major axis are (a, 0) and (?a, 0) and are referred to as the major intercepts or vertices. The endpoints of the minor axis are (0, b) and (0, -b) and are referred to as the minor intercepts or co-vertices. The points (c, 0) and (-c, 0) are the locations of the foci, and c can be found using the equation c2 = a2 - b2 . The key to understanding ellipses is understanding the three constants a, b and c.
Vertical
If an ellipse has its major axis along the y-axis and is centered at (0, 0), the standard form becomes:
The endpoints of the major axis become (0, a) and (0, -a). The endpoints of the minor axis are (b, 0) and ( ?b, 0). The foci are at (0, c) and (0, -c), with c2 = a2 - b2 .
When an ellipse is written in standard form, the major axis direction is determined by noting which variable has the larger denominator. The major axis either lies along that variable's axis or is parallel to that variable's axis.
Example 1
Graph the following ellipse. Find its vertices, length of the major axis, co-vertices, length of the minor axis, and foci.
Step 1 . Identify the vertices and length of major axis.
Since the larger denominator is with the y variable, the major axis lies along the y-axis. Since a2 = 9, |a| = 3.
Vertices: (0, 3), (0, -3)
Length of major axis: 6
Step 2. Identify the co-vertices and length of minor axis.
Since the smaller denominator is with the x variable, the minor axis lies along the x-axis. Since b2 = 4, |b| = 2.
Co-vertices: (2, 0), (-2, 0)
Length of minor axis: 4
Step 3. Identify the foci.
Step 4. Graph the ellipse.
Example 2
Graph the following ellipse. Find its vertices, co-vertices and foci.
Step 1 . Write the equation in standard form.
Divide each side of the equation by 100 and simplify.
Step 2. Identify the vertices, co-vertices and foci.
Since the larger denominator is with the x variable, the major axis lies along the x-axis.
Step 3. Graph the ellipse.
Center (h, k)
The standard form of an ellipse centered at (h, k) with its major axis parallel to the x-axis is 
.
The standard form for an ellipse centered at (h, k) with a major axis parallel to the y-axis is 
.
The rules of translation also apply to conic sections. For instance, if x is replaced with x ? h, the ellipse moves to the right by h. Use the generalizations below to determine the properties of an ellipse.
Example 3
Graph the following ellipse. Find its center, vertices, co-vertices and foci.
Step 1 . Find the center.
Center: (2, -1)
Step 2. Identify the vertices, co-vertices, and foci.
Since the larger denominator is with the x variable, the major axis lies along the x-axis.
Step 3. Graph the ellipse.
Example 4
Graph the following equation.
16x2 + 25y2 + 32x - 150y = 159
Step 1 . Identify the conic section.
This is an ellipse because the x and y terms are both squared,
have different coefficients, and are added.
Step 2 . Write equation in standard form.
(16x2+32x) + (25y2-150y) = 159; Group the x terms, group the y terms
16(x2+2x) + 25(y2-6y) = 159; Factor the leading coefficient from each group
16(x2+2x+1) + 25(y2-6y+9) = 159 + (16?1) + (25?9); Complete the square for x and y
16(x+1)2 + 25(y-3)2 = 400; Simplify
; Divide each side of the equation by 400 and simplify.
Step 3. Graph the conic.
Identify the center (-1, 3)
Identify the vertices (4, 3), (-6, 3)
Identify the co-vertices (-1, 7), (-1, -1)
Symmetry
Ellipses have both reflectional and rotational symmetry. The reflectional symmetry of every ellipse is a vertical line through the center and a horizontal line through the center. The rotational symmetry of every ellipse is 180 degrees about the center point.Example State the symmetry for the ellipse modeled by the equation:
Video Lessons
To learn how to graph ellipses, select the following link:
To learn how to write the equation of an ellipse, select the following link:
Use Standard Form to Write an Equations for an Ellipse
Interactive Activities
Compare the equation of an ellipse to its graph.
Guided Practice
To solidify your understanding of graphing and writing equations of ellipses, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Practice
*Note: If Google Docs displays, "Sorry, we were unable to retrieve the document for viewing," refresh your browser.
Video Lessons
To see more examples on circles, view the videos below.
Graphing an Ellipse in Standard FormÂ
Â
Graphing an Ellipse
-
Answer Key
*Note: If Google Docs displays, "Sorry, we were unable to retrieve the document for viewing," refresh your browser.
Sources
Sources used in this book: CliffsNotes.com. Ellipse. 27 Jul 2010
Holt, Rinehart & Winston , "Conic Sections ." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch10_03_ homeworkhelp.html (accessed 7/27/2010).
Kenny Felder, "Conic Concepts -- Circles," Connexions, March 22, 2010, http://cnx.org/content/m18245/1.3/.