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    圆通过三点

    发布时间:2020-10-08 15:38:20 作者:冬青好 

    在此页面中,穿过三个点的圆将给出示例问题,如果给出了穿过该圆的三个点,则该问题将查找圆的方程。

    列1:

    找出通过点(0,1)(2,3)和(-2,5)的圆的方程。

    圆的一般方程

    x²+y²+ 2gx + 2fy + c = 0


    现在我们必须在上述等式圆中通

    20201008153744.png

    通过点(0,1)的圆的方程

                            0²+1²+ 2g(0)+ 2 f(1)+ c = 0

                                   1 + 0 + 2f + c = 0

                                           2f + c = -1 ----------(1)

    通过点(2,3)的圆的方程

                            2²+3²+ 2g(2)+ 2 f(3)+ c = 0

                             4 + 9 + 4g + 6f + c = 0

                                      4g + 6f + c = -13 ----------(2)

    通过点(-2,5)的圆的方程

                       (-2)²+5²+ 2g(-2)+ 2 f(5)+ c = 0

                             4 + 25-4g + 10f + c = 0

                                  -4g + 10f + c = -29 ----------(3)

    所以这三个方程是

                                           2f + c = -1 ----------(1)

                                      4g + 6f + c = -13 ----------(2)

                                  -4g + 10f + c = -29 ----------(3)

    通过求解这三个方程,我们可以获得f,g和c的值

    通过添加(2)和(3)

                                      4g + 6f + c = -13 ----------(2)

                                  -4g + 10f + c = -29 ----------(3)

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

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

    将(1)乘以2

                                        4f + 2c = -2  

    减去(5)

                                         16f + 2c = -42

                                          4f + 2c = -2

                                         (-)(-)(+)    

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

                                           12f = -40

                                               f = -40/12   

                                               f = -10/3

    将f = -10/3代入第一个方程式

                                     2(-10/3)+ c = -1

                                       -20/3 + c = -1

                                                   c = -1 + 20/3

                                                   c = 17/3

    在第二个等式中替换c = 17/3和f = -10/3

                  4克+ 6(-10/3)+ 17/3 = -13

                  4克-20 + 17/3 =-13

                      4克+(-60 + 17)/ 3 = -13

                         4克-43/3 = -13

                                 4克= -13 + 43/3

                                 4克=(-39 + 43)/ 3

                                  4克= 4/3

                                    g = 4 /(4x3)

                                    g = 1/3

    将g = 1/3 c = 17/3代入f = -10/3

                        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         

    因此,通过三个点的圆的方程  3x²+3y²+ 2x -20y + 17 = 0

    更新:20210423 104205     


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