Set the equation of the circle:

X 2+Y 2+DX+EY+F = 0 (general formula)

Its central coordinates are: (-D/2, -E/2)

Because: the circle passes through point A (1, 4) and point B (3, 2).

Substitute a and b to get:

1^2+4^2+d+4e+f=0-[ 1]

3^2+(-2)^2+3D-2E+F=0 - [2]

Then from [1][2]:

E=(D-2)/3，F=-(46+7D)/3

Because: A (1, 4) B (3, 2)

Then: straight line AB: Y-4 = [(-2-4)/(3-1)] (x-1)

(point oblique)

That is: 3x+y-7=0.

Also, the distance from the center of the circle to the straight line AB is the root sign 10.

Then there is the root number 10=|3D+(D-2)/3-7|/ root number [3 2+12].

Solution: D= (double solution)

Then: E=(D-2)/3=, F=-(D+46)/3=

Then substitute x 2+y 2+dx+ey+f = 0 to get the equation of the circle.

(double solution)