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Determine the equation of the circle graphed below. 12 11 10 9. (9, 4) 4 (6, 3). 1 `-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1, 1 2 3 4 5 6 7 8 9 10 11 12 -2 -3 -4 -5 -6 -7 3.

Determine the equation of the circle graphed below. 12 11 10 9. (9, 4) 4 (6, 3). 1 `-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1, 1 2 3 4 5 6 7 8 9 10 11 12 -2 -3 -4 -5 -6 -7 3.

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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#### Determine the equation of the circle graphed below. Below you will find a graph with a circle centered at the point (6, 3) and passing through the point (9, 4). The graph is on a coordinate plane with marked axes (x and y), extending from -12 to 12 on both the x- and y-axes. The circle, in this case, has the following notable points: - Center: (6, 3) - A point on the circle: (9, 4) Based on the standard equation of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is its radius. Answer: [Input Box] To determine the radius, you can use the distance formula between the center and any given point on the circle: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Applying this formula with: Center \((x_1, y_1) = (6, 3)\) Point on circle \((x_2, y_2) = (9, 4)\), \[ r = \sqrt{(9 - 6)^2 + (4 - 3)^2} \] \[ r = \sqrt{3^2 + 1^2} \] \[ r = \sqrt{9 + 1} \] \[ r = \sqrt{10} \] Thus, the radius squared, \(r^2\), is 10. Substitute \(h\), \(k\), and \(r^2\) into the standard equation: \[ (x - 6)^2 + (y - 3)^2 = 10 \] Thus, the equation of the circle is: \[ (x - 6)^2 + (y - 3)^2 = 10 \]
Transcribed Image Text:#### Determine the equation of the circle graphed below. Below you will find a graph with a circle centered at the point (6, 3) and passing through the point (9, 4). The graph is on a coordinate plane with marked axes (x and y), extending from -12 to 12 on both the x- and y-axes. The circle, in this case, has the following notable points: - Center: (6, 3) - A point on the circle: (9, 4) Based on the standard equation of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is its radius. Answer: [Input Box] To determine the radius, you can use the distance formula between the center and any given point on the circle: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Applying this formula with: Center \((x_1, y_1) = (6, 3)\) Point on circle \((x_2, y_2) = (9, 4)\), \[ r = \sqrt{(9 - 6)^2 + (4 - 3)^2} \] \[ r = \sqrt{3^2 + 1^2} \] \[ r = \sqrt{9 + 1} \] \[ r = \sqrt{10} \] Thus, the radius squared, \(r^2\), is 10. Substitute \(h\), \(k\), and \(r^2\) into the standard equation: \[ (x - 6)^2 + (y - 3)^2 = 10 \] Thus, the equation of the circle is: \[ (x - 6)^2 + (y - 3)^2 = 10 \]
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