Find m and n if C(m,n) is the center of the circle circumscribed to the triangle AOB. A(4,-2), B(2,4)

The formula for the circle is:

x^2 + y^2 + ax + by +c =0

We have the triangle AOB , then A ,O and B are on the circle, Then A(4,-2) , O(0,0) and B(2,4) should verify the equation:

First we will subtitute with O(0,0):

==> 0+0+0+0+c= 0

==> c=0

Now substitute with A(4,-2):

16 + 4 + 4a -2b = 0

20 + 4a -2b =0 ......(1)

Now substitute with B(2,4):

4+ 16 +2a +4b =0

20 + 2a +4b =0 .....(2)

Now let us multiply (1) with 2 and add to (2):

40 +8a -4b =0

20 + 2a +4b= 0

==> 60 + 10a =0

==> a= -60/10 = -6

==> 2b= 20 +4a= 20-24= -4

==> b= -2

Now substitute a and b in the equation:

x^2 + y^2 -6x -2y =0

x^2 -6x +y^2 -2y =0

Complete the squares:

(x-3)^2 + (y-1)^2 -9 -1 =0

(x-3)^2 + (y-1)^2 =10

Then the center C is (3,1).