A circle has a diameter with endpoints (-8, 2) and (-2, 6). What is the equation of the circle? O2=(x-5)2+(y+ 4) 2 O2=(x+3)2+(y- 4) 2 O2=(x-3)2+(y+ 4) 2 O2=(x+5)2 +(y-4) 2
A circle has a diameter with endpoints (-8, 2) and (-2, 6). What is the equation of the circle? O2=(x-5)2+(y+ 4) 2 O2=(x+3)2+(y- 4) 2 O2=(x-3)2+(y+ 4) 2 O2=(x+5)2 +(y-4) 2
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Odysseyware Educational Platform: Geometry**
**Assignment - 2. Exam**
**Attempt 1 of 1**
**Question:**
A circle has a diameter with endpoints (-8, 2) and (-2, 6). What is the equation of the circle?
**Options:**
O \( r^2 = (x - 5)^2 + (y + 4)^2 \)
O \( r^2 = (x + 3)^2 + (y - 4)^2 \)
O \( r^2 = (x + 5)^2 + (y - 4)^2 \)
**Buttons:**
- Next Question
- Ask for Help
**Explanation:**
The information and problem displayed pertain to finding the equation of a circle given specific points that define its diameter. Identifying such equations requires comprehension of the midpoint formula for deriving the center of the circle and usage of the distance formula for calculating the radius.
From the endpoints of the diameter, (-8, 2) and (-2, 6), the center (h, k) can be determined as the midpoint:
\[ h = \frac{-8 + (-2)}{2} = -5 \]
\[ k = \frac{2 + 6}{2} = 4 \]
Thus, the center is at (-5, 4).
Next, the radius can be determined using the distance formula, finding the distance between one of the endpoints and the center:
\[ r = \sqrt{(-5 + 8)^2 + (4 - 2)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]
Inserting the center and radius into the standard circle equation:
\[ (x + 5)^2 + (y - 4)^2 = 13 \]
Thus, the correct option should be:
\[ r^2 = (x + 5)^2 + (y - 4)^2 \]
Students should validate their understanding by properly substituting known values into the hallmark circle equation and confirming their math operations.
For additional help, click "Ask for Help" or proceed to the "Next Question."
---
**Note:** This transcription includes instructional details apt for learners and educators, and ensures clarity in understanding the geometric concepts needed for solving the given problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7739b2ff-50c8-4d4b-9a5d-c9045fbc918c%2F984fdaa7-2b34-491c-a1bf-38fc575d2fe2%2Fc295esx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Odysseyware Educational Platform: Geometry**
**Assignment - 2. Exam**
**Attempt 1 of 1**
**Question:**
A circle has a diameter with endpoints (-8, 2) and (-2, 6). What is the equation of the circle?
**Options:**
O \( r^2 = (x - 5)^2 + (y + 4)^2 \)
O \( r^2 = (x + 3)^2 + (y - 4)^2 \)
O \( r^2 = (x + 5)^2 + (y - 4)^2 \)
**Buttons:**
- Next Question
- Ask for Help
**Explanation:**
The information and problem displayed pertain to finding the equation of a circle given specific points that define its diameter. Identifying such equations requires comprehension of the midpoint formula for deriving the center of the circle and usage of the distance formula for calculating the radius.
From the endpoints of the diameter, (-8, 2) and (-2, 6), the center (h, k) can be determined as the midpoint:
\[ h = \frac{-8 + (-2)}{2} = -5 \]
\[ k = \frac{2 + 6}{2} = 4 \]
Thus, the center is at (-5, 4).
Next, the radius can be determined using the distance formula, finding the distance between one of the endpoints and the center:
\[ r = \sqrt{(-5 + 8)^2 + (4 - 2)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]
Inserting the center and radius into the standard circle equation:
\[ (x + 5)^2 + (y - 4)^2 = 13 \]
Thus, the correct option should be:
\[ r^2 = (x + 5)^2 + (y - 4)^2 \]
Students should validate their understanding by properly substituting known values into the hallmark circle equation and confirming their math operations.
For additional help, click "Ask for Help" or proceed to the "Next Question."
---
**Note:** This transcription includes instructional details apt for learners and educators, and ensures clarity in understanding the geometric concepts needed for solving the given problem.
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