The circle centered at (2,-1) and with radius 4 intersects the circle centered at (2,6) and with radius sqrt(18) at two points A and B. Find (AB)^2.
+130123 The equations for the circles are
(x - 2)^2 + ( y + 1)^2 = 16 (1)
(x -2)^2 + ( y - 6)^2 = 18 (2) subtract (1) from (2)
(y - 6)^2 - (y + 1)^2 = 2
y^2 - 12y + 36 - y^2 - 2y - 1 = 2
-14y + 35 = 2
-14y = -33
y = -33 /-14 = 33/14 this is the y coordinate for both points
To find the x coordinates
(x -2)^2 + (33/14 + 1)^2 = 16
(x - 2)^2 + ( 47/14)^2 =16
(x - 2)^2 = 16 - (47/14)^2
( x-2)^2 = 927 / 196 take both roots
x - 2 = sqrt (927/196) or x - 2 = -sqrt (927/16)
x = sqrt (927/16) + 2 or x = -sqrt (927 /196) + 2
AB = [ sqrt (927/16) + 2 ] - [ sqrt (927 / 196) + 2 ] = 2 sqrt (927 /196)
(AB)^2 = 4 ( 927 / 196) = 927 / 49
