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 the Equation of a Circle
**Problem Statement:**
14.) Using the equation of a circle \((x - 3)^2 + y^2 = 49\), find the radius and the center.
**Solution Step-by-Step:**
The given equation of a circle is \((x - 3)^2 + y^2 = 49\).
1. **Equation Form:**
The standard form for 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.
2. **Finding the Center:**
By comparing \((x - 3)^2 + y^2 = 49\) to the standard equation \((x - h)^2 + (y - k)^2 = r^2\), it is clear that:
- \(h = 3\)
- \(k = 0\)
Therefore, the center of the circle is \((3, 0)\).
3. **Finding the Radius:**
The term on the right side of the equation \(49\) corresponds to \(r^2\):
\[ r^2 = 49 \]
Solving for \(r\),
\[ r = \sqrt{49} \]
\[ r = 7 \]
Therefore, the radius of the circle is \(7\).
**Summary:**
- The center of the circle is \((3, 0)\).
- The radius of the circle is \(7\).
By understanding the standard form of the circle’s equation, you can easily identify the center and radius from the given equation. This information is foundational in geometry and helps to graph and analyze circles accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92094b20-9792-4363-b2eb-d85ac30bc889%2F48a6f98b-ba76-4e8c-a092-6529e51162d5%2F3ashi4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Equation of a Circle
**Problem Statement:**
14.) Using the equation of a circle \((x - 3)^2 + y^2 = 49\), find the radius and the center.
**Solution Step-by-Step:**
The given equation of a circle is \((x - 3)^2 + y^2 = 49\).
1. **Equation Form:**
The standard form for 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.
2. **Finding the Center:**
By comparing \((x - 3)^2 + y^2 = 49\) to the standard equation \((x - h)^2 + (y - k)^2 = r^2\), it is clear that:
- \(h = 3\)
- \(k = 0\)
Therefore, the center of the circle is \((3, 0)\).
3. **Finding the Radius:**
The term on the right side of the equation \(49\) corresponds to \(r^2\):
\[ r^2 = 49 \]
Solving for \(r\),
\[ r = \sqrt{49} \]
\[ r = 7 \]
Therefore, the radius of the circle is \(7\).
**Summary:**
- The center of the circle is \((3, 0)\).
- The radius of the circle is \(7\).
By understanding the standard form of the circle’s equation, you can easily identify the center and radius from the given equation. This information is foundational in geometry and helps to graph and analyze circles accurately.
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