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
Identify each part of the circle given its equation
![### Problem 12:
Given the equation of a circle:
\[ (x + 7)^2 + (y + 2)^2 = 108 \]
#### Questions to Answer:
1. **Find the Center of the Circle:**
- The general form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.
- In this given equation, \((x + 7)^2 + (y + 2)^2 = 108\), we can rewrite it as \((x - (-7))^2 + (y - (-2))^2 = 108\).
- Therefore, the center of the circle is \((-7, -2)\).
2. **Find the Area of the Circle:**
- In a circle's equation \((x - h)^2 + (y - k)^2 = r^2\), \(r^2 = 108\).
- To find the radius \(r\), we take the square root of 108:
\[
r = \sqrt{108} = \sqrt{4 \times 27} = 2\sqrt{27}
\]
- The area \(A\) of a circle is given by the formula \(A = \pi r^2\).
- Given \(r^2 = 108\), the area \(A\) is:
\[
A = \pi \times 108 = 108\pi
\]
Therefore:
- **Center:** \((-7, -2)\)
- **Area:** \(108\pi\) square units.
Note: This transcription would be formatted to match the educational website's style guide, including appropriate spacing and formatting for clarity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3337d92-7fe5-45e7-a6c1-8ff8abb5ae9e%2F89d9edfb-ca5d-44f3-adb5-036bfaf7d98c%2F996i5n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 12:
Given the equation of a circle:
\[ (x + 7)^2 + (y + 2)^2 = 108 \]
#### Questions to Answer:
1. **Find the Center of the Circle:**
- The general form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.
- In this given equation, \((x + 7)^2 + (y + 2)^2 = 108\), we can rewrite it as \((x - (-7))^2 + (y - (-2))^2 = 108\).
- Therefore, the center of the circle is \((-7, -2)\).
2. **Find the Area of the Circle:**
- In a circle's equation \((x - h)^2 + (y - k)^2 = r^2\), \(r^2 = 108\).
- To find the radius \(r\), we take the square root of 108:
\[
r = \sqrt{108} = \sqrt{4 \times 27} = 2\sqrt{27}
\]
- The area \(A\) of a circle is given by the formula \(A = \pi r^2\).
- Given \(r^2 = 108\), the area \(A\) is:
\[
A = \pi \times 108 = 108\pi
\]
Therefore:
- **Center:** \((-7, -2)\)
- **Area:** \(108\pi\) square units.
Note: This transcription would be formatted to match the educational website's style guide, including appropriate spacing and formatting for clarity.
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