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
![**Circle Equations with Distance Formula**
**Date and Time:** Jun 18, 2:13:44 AM
**Watch Help Video**
---
**Instructions**
*Determine the equation of the circle graphed below:*
The given graph contains a circle plotted on a Cartesian coordinate system.
**Description of Graph:**
- The graph features an x-axis (horizontal) and y-axis (vertical) both marked from -12 to 12.
- A circle is centered around the point (3, 4).
- The circle passes through the point (4, 9).
- Both points (3, 4) and (4, 9) are highlighted in yellow, with labels indicating their coordinates.
---
**Diagram Explanation:**
- **Point (3, 4):** This is the center of the circle.
- **Point (4, 9):** This is a point on the circumference of the circle.
To determine the equation of the circle, use the standard form of the circle equation: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
**Steps:**
1. **Identify the center:** From the graph, the center is \((3, 4)\).
2. **Calculate the radius:** Use the distance formula to find the radius \(r\) between the center \((3, 4)\) and point \((4, 9)\).
The distance formula is:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Plugging in the coordinates:
\[ r = \sqrt{(4 - 3)^2 + (9 - 4)^2} = \sqrt{1^2 + 5^2} = \sqrt{1+25} = \sqrt{26} \]
The radius \(r\) is \(\sqrt{26}\).
3. **Write the equation:**
\[ (x - 3)^2 + (y - 4)^2 = (\sqrt{26})^2 \]
\[ (x - 3)^2 + (y - 4)^2 = 26 \]
**Answer:**
\[
(x - 3)^2 + (y - 4)^2 = 26
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffadf8417-7709-4f35-b9d0-c9fc01ac3c0b%2Fc34b5124-2968-49a6-8a88-020c1bb06ffd%2F2hvv587_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Circle Equations with Distance Formula**
**Date and Time:** Jun 18, 2:13:44 AM
**Watch Help Video**
---
**Instructions**
*Determine the equation of the circle graphed below:*
The given graph contains a circle plotted on a Cartesian coordinate system.
**Description of Graph:**
- The graph features an x-axis (horizontal) and y-axis (vertical) both marked from -12 to 12.
- A circle is centered around the point (3, 4).
- The circle passes through the point (4, 9).
- Both points (3, 4) and (4, 9) are highlighted in yellow, with labels indicating their coordinates.
---
**Diagram Explanation:**
- **Point (3, 4):** This is the center of the circle.
- **Point (4, 9):** This is a point on the circumference of the circle.
To determine the equation of the circle, use the standard form of the circle equation: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
**Steps:**
1. **Identify the center:** From the graph, the center is \((3, 4)\).
2. **Calculate the radius:** Use the distance formula to find the radius \(r\) between the center \((3, 4)\) and point \((4, 9)\).
The distance formula is:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Plugging in the coordinates:
\[ r = \sqrt{(4 - 3)^2 + (9 - 4)^2} = \sqrt{1^2 + 5^2} = \sqrt{1+25} = \sqrt{26} \]
The radius \(r\) is \(\sqrt{26}\).
3. **Write the equation:**
\[ (x - 3)^2 + (y - 4)^2 = (\sqrt{26})^2 \]
\[ (x - 3)^2 + (y - 4)^2 = 26 \]
**Answer:**
\[
(x - 3)^2 + (y - 4)^2 = 26
\
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