Circle Passing Through Three points

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)

                             ----------------------

                                     16f + 2c = -42   ---------(5)

Multiplying (1) by 2

                                    4f + 2c = -2  

Subtract (5) form this equation

                                     16f + 2c = -42

                                      4f + 2c  = -2

                                     (-)   (-)     (+)    

                                   ----------------

                                       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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