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
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![**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
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