+4609 The circle centered at (2,-1) and with radius 4 intersects the circle centered at (2,5) and with radius \((2,-1)\) at two points A and B . Find\((AB)^2\) .
+128474 We have the following two equations
(x - 2)^2 + (y + 1)^2 = 16
(x - 2)^2 + (y - 5)^2 = 16
Setting the two equations equal and subtracting (x-2)^2 from both sides. we have that
(y + 1)^2 = (y - 5)^2
y^2 + 2y + 1 = y^2 - 10y + 25 rearrange
12y = 24
y = 2
And subbing this into either of the original equation we have that
(x - 2)^2 + 3^2 = 16
(x - 2)^2 = 7 take both roots
x - 2 = ±√7 add 2 to both sides
x = ±√7 + 2
So....the intersection points are A, B = (√7 + 2, 2) and ( -√7 + 2, 2)
My question, tertre, is how to interpret (AB)^2.......are we supposed to take the dot product of these points and square that???
If so..... [A (dot) B]^2 = [ 4 - 7 + 4]^2 = 1^2 = 1
