The x coordinates of both circles will be the same
Equation of first circle
(x -2)^2 + (y + 1)^2 = 16
Equation of second circle
(x -2)^2 + (y -5)^2 = 14
If we subtract the second equation from the first we get
(y + 1)^2 - (y -5)^2 = 2 simplify
y^2 + 2y + 1 - ( y^2 - 10y + 25) = 2
12y -24 = 2
12y =26
y = 26/12 = 13/6
Usig either equation to find the x values of intersection
(x - 2)^2 + (13/6 +1)^2 = 16
(x - 2)^2 + ( 19/6)^2 = 16
(x - 2)^2 = 16 - (19/6)^2
(x - 2)^2 = 215/36
x - 2 = sqrt (215)/6
x = sqrt (215) / 6 + 2 or x = -sqrt (215)/ 6 + 2
(AB)^2 is just the squared distance between A and B.......this is just the difference between the x coordinates (squared)
[ ( ( sqrt (215/6 + 2) - (-sqrt (215)/6 + 2) ) ]^2 =
[ 2 sqrt (215) / 6 ]^2 = 215 / 9 = (AB)^2
