Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![### Understanding and Writing the Equation of an Ellipse
**Graph Description:**
In the provided graph, we see a grid with both horizontal and vertical axes ranging from -9 to 9. The blue ellipse is centered at the origin (0,0). The ellipse stretches vertically from -4 to 4 and horizontally from -2 to 2, indicating a longer vertical axis.
**Ellipse Features:**
- **Center:** The center of the ellipse is at (0,0).
- **Major Axis:** The longer axis is the vertical one with endpoints at (0,4) and (0,-4), making the length of the semi-major axis 4.
- **Minor Axis:** The shorter axis is the horizontal one with endpoints at (2,0) and (-2,0), making the length of the semi-minor axis 2.
**Equation of the Ellipse:**
Given the center at (0,0), the equation of an ellipse in standard form is:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
Here, \(a\) is the semi-major axis and \(b\) is the semi-minor axis.
For this ellipse:
- \(a = 4\) (vertical)
- \(b = 2\) (horizontal)
Therefore, the equation of the ellipse is:
\[
\frac{x^2}{2^2} + \frac{y^2}{4^2} = 1
\]
Simplifying gives:
\[
\frac{x^2}{4} + \frac{y^2}{16} = 1
\]
**Task:**
Write an equation for the ellipse graphed in standard form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94589190-d7d8-41be-9c6e-d2d51b756765%2Fbcc382df-b9c4-4272-a00e-e426f6c03a47%2Fy3qepko_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding and Writing the Equation of an Ellipse
**Graph Description:**
In the provided graph, we see a grid with both horizontal and vertical axes ranging from -9 to 9. The blue ellipse is centered at the origin (0,0). The ellipse stretches vertically from -4 to 4 and horizontally from -2 to 2, indicating a longer vertical axis.
**Ellipse Features:**
- **Center:** The center of the ellipse is at (0,0).
- **Major Axis:** The longer axis is the vertical one with endpoints at (0,4) and (0,-4), making the length of the semi-major axis 4.
- **Minor Axis:** The shorter axis is the horizontal one with endpoints at (2,0) and (-2,0), making the length of the semi-minor axis 2.
**Equation of the Ellipse:**
Given the center at (0,0), the equation of an ellipse in standard form is:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
Here, \(a\) is the semi-major axis and \(b\) is the semi-minor axis.
For this ellipse:
- \(a = 4\) (vertical)
- \(b = 2\) (horizontal)
Therefore, the equation of the ellipse is:
\[
\frac{x^2}{2^2} + \frac{y^2}{4^2} = 1
\]
Simplifying gives:
\[
\frac{x^2}{4} + \frac{y^2}{16} = 1
\]
**Task:**
Write an equation for the ellipse graphed in standard form.
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