Questio Write the equation for a length of 10 units and a full minor x-axis length of 4 units. +==1 16 25 A B U D 25 9 x2 9 25 1 = 1 vertically-oriented ellipse centered at (0,0) with a full major y-axis All Changes Saved
Questio Write the equation for a length of 10 units and a full minor x-axis length of 4 units. +==1 16 25 A B U D 25 9 x2 9 25 1 = 1 vertically-oriented ellipse centered at (0,0) with a full major y-axis All Changes Saved
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Question 7
**Prompt:** Write the equation for a vertically-oriented ellipse centered at (0,0) with a full major y-axis length of 10 units and a full minor x-axis length of 4 units.
The following options are provided for the equation of the ellipse:
**A.** \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \)
**B.** \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \)
**C.** \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \)
**D.** \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \)
**Explanation:**
An ellipse centered at (0,0) with a major axis along the y-axis and a minor axis along the x-axis can be written in the standard form:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
Where:
- \( a \) is the semi-major axis length.
- \( b \) is the semi-minor axis length.
Given that the full major y-axis length is 10 units, the semi-major axis length \( a \) is 5 units (since \( a = \frac{10}{2} \)).
Similarly, the full minor x-axis length is 4 units, then the semi-minor axis length \( b \) is 2 units (\( b = \frac{4}{2} \)).
Therefore, the equation of the ellipse is:
\[ \frac{x^2}{2^2} + \frac{y^2}{5^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{25} = 1 \]
Hence, the correct option is **C.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8182640d-a39f-4906-bb1d-1d29c4c23cc2%2F5d41872a-582e-4267-9c8b-1f5259a915f7%2Fpc8uaar_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 7
**Prompt:** Write the equation for a vertically-oriented ellipse centered at (0,0) with a full major y-axis length of 10 units and a full minor x-axis length of 4 units.
The following options are provided for the equation of the ellipse:
**A.** \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \)
**B.** \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \)
**C.** \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \)
**D.** \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \)
**Explanation:**
An ellipse centered at (0,0) with a major axis along the y-axis and a minor axis along the x-axis can be written in the standard form:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
Where:
- \( a \) is the semi-major axis length.
- \( b \) is the semi-minor axis length.
Given that the full major y-axis length is 10 units, the semi-major axis length \( a \) is 5 units (since \( a = \frac{10}{2} \)).
Similarly, the full minor x-axis length is 4 units, then the semi-minor axis length \( b \) is 2 units (\( b = \frac{4}{2} \)).
Therefore, the equation of the ellipse is:
\[ \frac{x^2}{2^2} + \frac{y^2}{5^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{25} = 1 \]
Hence, the correct option is **C.**
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