Given the points (2, 16), (11, 13), and (14, 10). Set up the general equation for a circle, of radius R, centered at (A, B): (x- A)^2 + (y - B)^2 = R^2 Construct 3 copies of this equation, each with the three points substituted for x and y: (2 - A)^2 + (16 - B)^2 = R^2 (11- A)^2 + (13 - B)^2 = R^2 (14- A)^2 + (10 - B)^2 = R^2 Eliminate R^2 by equating them to each other: (11- A)^2 + (13 - B)^2 = (2 - A)^2 + (16 - B)^2 (14- A)^2 + (10 - B)^2 = (2 - A)^2 + (16 - B)^2 Expand: A^2 - 4*A + B^2 - 32*B + 260 = A^2 - 22*A + B^2 - 26 B + 290 A^2 - 4*A + B^2 - 32*B + 260 = A^2 - 28*A + B^2 - 20 B + 296 Gather the unknown terms to the left and known numbers to the right. Notice that A^2 and B^2 both cancel, so this reduces from a quadratic system to a linear system. 18*A - 6*B = 30 24*A - 12*B = 36 Solve for A&B: A = 2, B = 1 Pick an equation and substitute known A and B to find R: (14- 2)^2 + (10 - 1)^2 = R^2 R^2 = 225 R = 15 Conclusion: Center at (2, 1), radius R = 15
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Given 3 points on a circle, find the center (and the radius). Geometrically, then algebraically.
Given the points (2, 16), (11, 13), and (14, 10).
Set up the general equation for a circle, of radius R, centered at (A, B):
(x- A)^2 + (y - B)^2 = R^2
Construct 3 copies of this equation, each with the three points substituted for x and y:
(2 - A)^2 + (16 - B)^2 = R^2
(11- A)^2 + (13 - B)^2 = R^2
(14- A)^2 + (10 - B)^2 = R^2
Eliminate R^2 by equating them to each other:
(11- A)^2 + (13 - B)^2 = (2 - A)^2 + (16 - B)^2
(14- A)^2 + (10 - B)^2 = (2 - A)^2 + (16 - B)^2
Expand:
A^2 - 4*A + B^2 - 32*B + 260 = A^2 - 22*A + B^2 - 26 B + 290
A^2 - 4*A + B^2 - 32*B + 260 = A^2 - 28*A + B^2 - 20 B + 296
Gather the unknown terms to the left and known numbers to the right. Notice that A^2 and B^2 both cancel, so this reduces from a quadratic system to a linear system.
18*A - 6*B = 30
24*A - 12*B = 36
Solve for A&B:
A = 2, B = 1
Pick an equation and substitute known A and B to find R:
(14- 2)^2 + (10 - 1)^2 = R^2
R^2 = 225
R = 15
Conclusion:
Center at (2, 1), radius R = 15
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01:20 I was waiting for the parentheses around the squared -2, and then there they were.
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