The circle centered at (2,-1) and with radius 4 intersects the circle centered at (2,5) and with radius sqrt(17) at two points A and B. Find (AB)^2.
first circle:
(x-2)2 + (y+1)2 = 16
becomes:
(x2 - 4x + 4) + (y2 +2y + 1) = 16
x2 - 4x + y2 + 2y + 5 = 16
x2 - 4x + y2 + 2y = 11
second circle:
(x-2)2 + (y-5)2 = 17
becomes:
x2 - 4x + 4 + y2 - 10y + 25 = 17
x2 - 4x + y2 - 10y + 29 = 17
x2 - 4x + y2 - 10y + 23 = 11
we have x2 - 4x + y2 + 2y = 11 and x2 - 4x + y2 - 10y + 23 = 11
both equal 11 so they equal each other
x2 - 4x + y2 + 2y = x2 - 4x + y2 - 10y + 23
x2, -4x, y2 cancel out
2y = -10y +23
12y = 23
y = 23/12
23/12 is the y - value of both of the points where the circles intersect (A and B)
choose one of the equations to plug in 23/12 as y. This will find the x-values of both of the points
i choose (x-2)2 + (y+1)2 = 16
but you can do other one too
(x-2)2 + ((23/12)+1)2 = 16
(x-2)2 + ((23/12)+(12/12))2 = 16
(x-2)2 + (35/12)2 = 16
(x-2)2 + 1225/144 = 2304/144
(x-2)2 = 1079/144
(x-2) = sqrt(1079/144) OR (x-2) = -sqrt(1079/144)
x = sqrt(1079/144) + 2 OR -sqrt(1079/144) + 2
These are the x values of the two points where the circle intersect
I guess AB means the line segment between them
So AB2 is the length of the line segment squared which means we subtract the two x-values then square. We dont have to worry about the y-values since they are the same for both points
( (sqrt(1079/144) + 2) - (-sqrt(1079/144) + 2) )2
( (sqrt(1079/144) + 2 + sqrt(1079/144) - 2) )2
( sqrt(1079/144) + sqrt(1079/144) )2
( 2sqrt(1079/144) )2
4(1079/144)
1079/36
or in decimal
29.97222....