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
![**5) Write the standard equation of the circle.**
In this problem, you are asked to determine the standard equation of a circle based on the provided diagram.
**Diagram Description:**
- The diagram shows a grid with a circle centered at the origin (0, 0).
- The circle intersects the x-axis and y-axis at points (1, 0), (-1, 0), (0, 1), and (0, -1).
- The grid is labeled with units, and the marks on the axes go from -3 to 3, showing a uniform scale.
**Detailed Explanation:**
1. **Center of the Circle:**
- The center of the circle is at the origin (0, 0).
2. **Radius of the Circle:**
- The circle intersects the x-axis and y-axis at a distance of 1 unit from the center.
- Therefore, the radius \( r \) of the circle is 1 unit.
3. **Standard Equation of a Circle:**
- The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
4. **Substitute the Center and Radius:**
- In this case, the center \((h, k)\) is \((0, 0)\), and the radius \( r \) is 1.
- Substituting these values into the standard form equation, we get:
\[
(x - 0)^2 + (y - 0)^2 = 1^2
\]
Which simplifies to:
\[
x^2 + y^2 = 1
\]
So, the standard equation of the circle is:
\[
x^2 + y^2 = 1
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42779bdc-66e8-47e8-8a59-0cd69b39b8c3%2F79ae37d8-d001-4cee-bf23-efe010d8b175%2Ftrg7k74_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**5) Write the standard equation of the circle.**
In this problem, you are asked to determine the standard equation of a circle based on the provided diagram.
**Diagram Description:**
- The diagram shows a grid with a circle centered at the origin (0, 0).
- The circle intersects the x-axis and y-axis at points (1, 0), (-1, 0), (0, 1), and (0, -1).
- The grid is labeled with units, and the marks on the axes go from -3 to 3, showing a uniform scale.
**Detailed Explanation:**
1. **Center of the Circle:**
- The center of the circle is at the origin (0, 0).
2. **Radius of the Circle:**
- The circle intersects the x-axis and y-axis at a distance of 1 unit from the center.
- Therefore, the radius \( r \) of the circle is 1 unit.
3. **Standard Equation of a Circle:**
- The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
4. **Substitute the Center and Radius:**
- In this case, the center \((h, k)\) is \((0, 0)\), and the radius \( r \) is 1.
- Substituting these values into the standard form equation, we get:
\[
(x - 0)^2 + (y - 0)^2 = 1^2
\]
Which simplifies to:
\[
x^2 + y^2 = 1
\]
So, the standard equation of the circle is:
\[
x^2 + y^2 = 1
\]
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