Question 207637: Circle U passes through points A(6,8), B(16, -4) and C(10, 8). Find the coordinates of the center of the circle. Describe in detail the method you use.
Thanks for the help...
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! Circle U passes through points A(6,8), B(16, -4) and C(10, 8). Find the coordinates of the center of the circle. Describe in detail the method you use.
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There's more than one way to do this.
I do it by using an Excel sheet I made that gives the equation of the circle using matrices.
|s..x..y..1|
|s1 x1 y1 1|
|s2 x2 y2 1| = 0
|s3 x3 y3 1|
s is the sum of x^2 + y^2
Solving that matrix gives the equation of the circle, in this case it's
x^2 + y^2 - 16x + y = 12
Then complete the squares
x^2 - 16x + 64 + y^2 + y + 1/4 = 12 + 64 + 1/4 = 76.25
(x-8)^2 + (y+0.5)^2 = 76.25
That puts the center at (8,-1/2)
This is not the quickest way, but I already have the Excel sheet that does it.
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Another method, and probably the one expected, is to find the perpendicular bisectors of 2 of the lines between pairs of points given, then solve for where they intersect. That's the center of the circle.
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points A(6,8), B(16,-4) and C(10,8)
1. Find the midpoint of AC
AC is parallel to the x-axis (both y values are 8) and its slope is zero.
The midpoint x value is (6+10)/2 = 8. That gives the point (8,8), and it's a vertical line. The center is on that line.
Its equation is x = 8
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2. Find the midpoint of AB
Find the x and y values separately by getting the average:
for x: (6+16)/2 = 11
for y: (8-4)/2 = 2
--> (11,2)
Find the slope, m, of AB
m = diffy/diffx = (-4-8)/(16-6)
m = -6/5
Find the equation of the line perpendicular to AB thru (11,2)
y-y1 = m*(x-x1) (m = negative inverse of -6/5)
y-2 = (5/6)*(x-11)
6y-12 = 5*(x-11)
6y-12 = 5x-55
6y = 5x-43
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Solve 6y = 5x-43 and x = 8 for x and y.
6y = 5*8 - 43 = -3
y = -1/2
--> (8,-1/2) same answer.
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