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
100%
![**Finding the Equation of a Circle**
**Problem Statement:**
C is the center of the circle shown below. What is the equation of circle C?
**Diagram Description:**
The diagram shows a circle with center labeled as C. There are two points marked on the circumference of the circle: (-3, 4) and (0, 2).
**Solution:**
To find the equation of the circle, we need to determine the center and the radius 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 its radius.
1. **Determine the Center (h, k):**
From the diagram, it is given that C is the center of the circle. The coordinates for the center are \((0, 2)\).
2. **Calculate the Radius (r):**
The radius can be found using the distance formula between the center and any point on the circle. We will use the point (-3, 4).
The distance formula is:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting in the points \((0, 2)\) and \((-3, 4)\):
\[
r = \sqrt{(0 + 3)^2 + (2 - 4)^2}
\]
\[
r = \sqrt{3^2 + (-2)^2}
\]
\[
r = \sqrt{9 + 4}
\]
\[
r = \sqrt{13}
\]
3. **Form the Equation:**
Now, substituting the center \((0, 2)\) and radius \(r = \sqrt{13}\) into the general equation of the circle, we get:
\[
(x - 0)^2 + (y - 2)^2 = (\sqrt{13})^2
\]
Simplifying:
\[
x^2 + (y - 2)^2 = 13
\]
**Equation of the Circle:**
\[
x^2 + (y - 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3f0117e-26e1-4209-8b43-253a6b092b43%2F92be952b-2fc5-40c5-bcd1-ac27ba37b1a2%2Fjcm83bk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Equation of a Circle**
**Problem Statement:**
C is the center of the circle shown below. What is the equation of circle C?
**Diagram Description:**
The diagram shows a circle with center labeled as C. There are two points marked on the circumference of the circle: (-3, 4) and (0, 2).
**Solution:**
To find the equation of the circle, we need to determine the center and the radius 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 its radius.
1. **Determine the Center (h, k):**
From the diagram, it is given that C is the center of the circle. The coordinates for the center are \((0, 2)\).
2. **Calculate the Radius (r):**
The radius can be found using the distance formula between the center and any point on the circle. We will use the point (-3, 4).
The distance formula is:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting in the points \((0, 2)\) and \((-3, 4)\):
\[
r = \sqrt{(0 + 3)^2 + (2 - 4)^2}
\]
\[
r = \sqrt{3^2 + (-2)^2}
\]
\[
r = \sqrt{9 + 4}
\]
\[
r = \sqrt{13}
\]
3. **Form the Equation:**
Now, substituting the center \((0, 2)\) and radius \(r = \sqrt{13}\) into the general equation of the circle, we get:
\[
(x - 0)^2 + (y - 2)^2 = (\sqrt{13})^2
\]
Simplifying:
\[
x^2 + (y - 2)^2 = 13
\]
**Equation of the Circle:**
\[
x^2 + (y - 2
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning