Determine the equation of the circle graphed below. 12 11 10 6. 8 7. 6. -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 89 10 11 12 (-1, -2) -3 -4 (-3: -5). -5 -6 -10 -11 -12 543 2 1,
Determine the equation of the circle graphed below. 12 11 10 6. 8 7. 6. -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 89 10 11 12 (-1, -2) -3 -4 (-3: -5). -5 -6 -10 -11 -12 543 2 1,
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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Question
![**Title: Determine the Equation of the Circle Graphed Below**
**Instructions:**
To find the equation of the circle, follow these steps:
1. **Identify the center of the circle (h, k):**
- The center of the circle is the point where the circle is perfectly centered.
- From the graph, the center (-3, -5) is indicated clearly.
2. **Determine the radius r:**
- The radius is the distance from the center of the circle to any point on the circle.
- One point on the circle shown in the graph is (-1, -2).
- Use the distance formula to calculate the radius:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \((x_1, y_1) = (-3, -5)\) and \((x_2, y_2) = (-1, -2)\):
\[
r = \sqrt{(-1 - (-3))^2 + (-2 - (-5))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{13}
\]
3. **Form the equation of the circle:**
- The standard form for the equation of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- Plug in the values for \(h\), \(k\), and \(r\):
\[
(x + 3)^2 + (y + 5)^2 = 13
\]
**Graph Description:**
- **Axes:**
- The x-axis and y-axis range from -12 to 12.
- The origin (0,0) is clearly marked.
- **Grid:**
- The graph is overlaid with a grid pattern for precise plotting.
- **Circle:**
- A circle is drawn with the center at \((-3, -5)\).
- The circumference of the circle passes through the point \((-1, -2)\).
By following these steps, you can determine that the equation of the circle graphed is:
\[
(x + 3)^2 + (y + 5)^2 = 13
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03ee4b44-fd9b-4d43-bc23-25b15a4cdf81%2F376db204-6213-4532-84fd-a72e109580b2%2Fvjqsk8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determine the Equation of the Circle Graphed Below**
**Instructions:**
To find the equation of the circle, follow these steps:
1. **Identify the center of the circle (h, k):**
- The center of the circle is the point where the circle is perfectly centered.
- From the graph, the center (-3, -5) is indicated clearly.
2. **Determine the radius r:**
- The radius is the distance from the center of the circle to any point on the circle.
- One point on the circle shown in the graph is (-1, -2).
- Use the distance formula to calculate the radius:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \((x_1, y_1) = (-3, -5)\) and \((x_2, y_2) = (-1, -2)\):
\[
r = \sqrt{(-1 - (-3))^2 + (-2 - (-5))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{13}
\]
3. **Form the equation of the circle:**
- The standard form for the equation of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- Plug in the values for \(h\), \(k\), and \(r\):
\[
(x + 3)^2 + (y + 5)^2 = 13
\]
**Graph Description:**
- **Axes:**
- The x-axis and y-axis range from -12 to 12.
- The origin (0,0) is clearly marked.
- **Grid:**
- The graph is overlaid with a grid pattern for precise plotting.
- **Circle:**
- A circle is drawn with the center at \((-3, -5)\).
- The circumference of the circle passes through the point \((-1, -2)\).
By following these steps, you can determine that the equation of the circle graphed is:
\[
(x + 3)^2 + (y + 5)^2 = 13
\]
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