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Use the graph below to determine the equation of the circle in (a) center-radius form and (b) general form. a. Type the equation in center-radius form. A 10+ (Simplify your answer.) -10 to

Use the graph below to determine the equation of the circle in (a) center-radius form and (b) general form. a. Type the equation in center-radius form. A 10+ (Simplify your answer.) -10 to

Algebra and Trigonometry (6th Edition)
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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**Determining the Equation of a Circle: Center-Radius Form and General Form** ### Overview This educational module will guide you through the process of determining the equation of a circle using a graph. You will learn how to express the circle's equation in both center-radius form and general form. ### Task Use the graph provided to determine the equation of the circle in: 1. Center-radius form 2. General form ### Graph Description The graph consists of a coordinate plane with a circle plotted on it. Key points on the circle are labeled: - The center of the circle is at point (4, 1). - Points on the circumference are approximately located at (4, 7), (1, 1), and (7, 1). #### Steps to Follow 1. **Center-Radius Form**: - Identify the center of the circle (h, k) from the graph. Here, it is (4, 1). - Determine the radius of the circle. The distance between the center and any point on the circumference (for example, from (4, 1) to (4, 7)) is the radius. The radius here is 6. - The center-radius form of the equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] - Substitute (h, k) = (4, 1) and r = 6 into the equation: \[ (x - 4)^2 + (y - 1)^2 = 6^2 \] - Simplify the equation to: \[ (x - 4)^2 + (y - 1)^2 = 36 \] 2. **General Form**: - Expand the center-radius form equation: \[ (x - 4)^2 + (y - 1)^2 = 36 \] - Expand (x - 4)^2 and (y - 1)^2 to: \[ x^2 - 8x + 16 + y^2 - 2y + 1 = 36 \] - Combine like terms to obtain the general form of the equation: \[ x^2 + y^2 - 8x - 2y + 17
Transcribed Image Text:**Determining the Equation of a Circle: Center-Radius Form and General Form** ### Overview This educational module will guide you through the process of determining the equation of a circle using a graph. You will learn how to express the circle's equation in both center-radius form and general form. ### Task Use the graph provided to determine the equation of the circle in: 1. Center-radius form 2. General form ### Graph Description The graph consists of a coordinate plane with a circle plotted on it. Key points on the circle are labeled: - The center of the circle is at point (4, 1). - Points on the circumference are approximately located at (4, 7), (1, 1), and (7, 1). #### Steps to Follow 1. **Center-Radius Form**: - Identify the center of the circle (h, k) from the graph. Here, it is (4, 1). - Determine the radius of the circle. The distance between the center and any point on the circumference (for example, from (4, 1) to (4, 7)) is the radius. The radius here is 6. - The center-radius form of the equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] - Substitute (h, k) = (4, 1) and r = 6 into the equation: \[ (x - 4)^2 + (y - 1)^2 = 6^2 \] - Simplify the equation to: \[ (x - 4)^2 + (y - 1)^2 = 36 \] 2. **General Form**: - Expand the center-radius form equation: \[ (x - 4)^2 + (y - 1)^2 = 36 \] - Expand (x - 4)^2 and (y - 1)^2 to: \[ x^2 - 8x + 16 + y^2 - 2y + 1 = 36 \] - Combine like terms to obtain the general form of the equation: \[ x^2 + y^2 - 8x - 2y + 17
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