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Circle Passing Through Three Given Points

EXAMPLE 5:

Find the equation of the circle passing through the points

( 9 , -7 ) , ( - 3 , -1 ) and ( 6 , 2 ) .

2
2
Suppose the equation of the circle is ( 9 , -7 ) : 9 + (-7) + 2g(9) + 2f(-7) + c = 0
18 g - 14 f + c = -130 1

2
x + y + 2gx + 2fy + c = 0
2
2
2
( - 3 , -1 ) : (-3) + (-1) + 2g(-3) + 2f(-1) + c = 0
- 6g - 2f + c = -10 2
Substituting the coordinates of each of the
three points into this equation gives: ( 6 , 2 ) : 6 + 2 + 2g(6) + 2f(2) + c = 0
2
2
12 g + 4 f + c = - 40 3