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Determine the equation of the circle graphed below. 10 8 -10 -8 -6 -4 -2 8 10 -2 -6 -8 2. 6 4. 2.

Determine the equation of the circle graphed below. 10 8 -10 -8 -6 -4 -2 8 10 -2 -6 -8 2. 6 4. 2.

Elementary Geometry For College Students, 7e
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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### Determining the Equation of a Circle **Problem Statement:** Determine the equation of the circle graphed below. **Graph Explanation:** The given graph is a standard Cartesian coordinate plane with both x and y-axes ranging from -10 to 10. The circle on the graph has a center at (4, -2) and appears to intersect the x-axis at points (1, -2) and (7, -2). **Steps to Determine the Equation:** 1. **Identify the Center:** The center of the circle is at point (4, -2). 2. **Calculate the Radius:** To determine the radius, use the distance formula between the center and any point on the circle. In this case, the circle intersects the x-axis at (7, -2), which is 3 units away from the center. \[ r = \text{Distance between (4, -2) and (7, -2)} = |7 - 4| = 3 \] 3. **Formulate the Equation:** Using the standard form equation of a circle \[ (x - h)^2 + (y - k)^2 = r^2 \] where (h, k) is the center and r is the radius: \[ (x - 4)^2 + (y + 2)^2 = 3^2 \] Simplifying this, we get: \[ (x - 4)^2 + (y + 2)^2 = 9 \] **Final Equation:** The equation of the circle is: \[ (x - 4)^2 + (y + 2)^2 = 9 \] Feel free to interact with the graph to better understand the properties of circles in coordinate geometry. For more practice problems and interactive content, visit our Geometry section.
Transcribed Image Text:### Determining the Equation of a Circle **Problem Statement:** Determine the equation of the circle graphed below. **Graph Explanation:** The given graph is a standard Cartesian coordinate plane with both x and y-axes ranging from -10 to 10. The circle on the graph has a center at (4, -2) and appears to intersect the x-axis at points (1, -2) and (7, -2). **Steps to Determine the Equation:** 1. **Identify the Center:** The center of the circle is at point (4, -2). 2. **Calculate the Radius:** To determine the radius, use the distance formula between the center and any point on the circle. In this case, the circle intersects the x-axis at (7, -2), which is 3 units away from the center. \[ r = \text{Distance between (4, -2) and (7, -2)} = |7 - 4| = 3 \] 3. **Formulate the Equation:** Using the standard form equation of a circle \[ (x - h)^2 + (y - k)^2 = r^2 \] where (h, k) is the center and r is the radius: \[ (x - 4)^2 + (y + 2)^2 = 3^2 \] Simplifying this, we get: \[ (x - 4)^2 + (y + 2)^2 = 9 \] **Final Equation:** The equation of the circle is: \[ (x - 4)^2 + (y + 2)^2 = 9 \] Feel free to interact with the graph to better understand the properties of circles in coordinate geometry. For more practice problems and interactive content, visit our Geometry section.
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