In this page circle passing through three points we are going to see example problems to find the equation of a circle if three points passing through the circle is given.
Example 1:
Find the equation of a circle passing through the points (0,1) (2,3) and (-2,5).
The general equation of a circle
x² + y² + 2gx + 2fy + c = 0
Now we have to apply these points one by one in the above equation circle passing through three points
equation of a circle which is passing through the point (0,1)
0² + 1² + 2g(0) + 2 f(1) + c = 0
1 + 0 + 2f + c = 0
2f + c = -1 ----------(1)
equation of a circle which is passing through the point (2,3)
2² + 3² + 2g(2) + 2 f(3) + c = 0
4 + 9 + 4g + 6f + c = 0
4g + 6f + c = -13 ----------(2)
equation of a circle which is passing through the point (-2,5)
(-2)² + 5² + 2g(-2) + 2 f(5) + c = 0
4 + 25 - 4g + 10f + c = 0
-4g + 10f + c = -29 ----------(3)
So the three equations are
2f + c = -1 ----------(1)
4g + 6f + c = -13 ----------(2)
-4g + 10f + c = -29 ----------(3)
by solving these three equations we can get the values of f,g and c
by adding (2) and (3)
4g + 6f + c = -13 ----------(2)
-4g + 10f + c = -29 ----------(3)
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16f + 2c = -42 ---------(5)
Multiplying (1) by 2
4f + 2c = -2
Subtract (5) form this equation
16f + 2c = -42
4f + 2c = -2
(-) (-) (+)
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12f = -40
f = -40/12
f = -10/3
Substitute f = -10/3 in the first equation
2(-10/3) + c = -1
-20/3 + c = -1
c = -1 + 20/3
c = 17/3
substitute c = 17/3 and f = -10/3 in the second equation
4g + 6(-10/3) + 17/3 = -13
4g - 20 + 17/3 = - 13
4g + (-60+17)/3 = -13
4g - 43/3 = -13
4g = -13 + 43/3
4g = (-39+43)/3
4g = 4/3
g = 4/(4x3)
g = 1/3
Substitute g = 1/3 c = 17/3 ad f = -10/3 in the general equation
x² + y² + 2gx + 2fy + c = 0
x² + y² + 2(1/3)x + 2(-10/3)y + 17/3 = 0
x² + y² + (2/3)x + (-20/3)y + 17/3 = 0
3x² + 3y² + 2x -20y + 17 = 0
So the equation of a circle passing through three points 3x² + 3y² + 2x -20y + 17 = 0
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