Circle A has a radius of #2 # and a center of #(8 ,2 )#. Circle B has a radius of #4 # and a center of #(2 ,3 )#. If circle B is translated by #<-1 ,5 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Under a translation of # ((-1),(5))#

We now require to calculate the distance between the centres of A and B using the #color(blue)" distance formula "#

# d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) #

where #(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points "#

let # (x_1,y_1)=(8,2)" and " (x_2,y_2)=(1,8)#

# d = sqrt((1-8)^2 + (8-2)^2) = sqrt(49+36) = sqrt85 ≈ 9.22 #

To determine if Circle B overlaps Circle A after being translated by <-1, 5>, we need to find the distance between the centers of the circles and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, the circles overlap; otherwise, they do not. First, let's find the new center of Circle B after translation: Original center of Circle B: (2, 3) Translation vector: <-1, 5> New center of Circle B: (2 - 1, 3 + 5) = (1, 8) Now, let's find the distance between the centers of the circles: Distance between centers = √((x2 - x1)^2 + (y2 - y1)^2) = √((8 - 1)^2 + (2 - 8)^2) = √(49 + 36) = √85 Now, let's compare the distance between the centers to the sum of the radii: Radius of Circle A: 2 Radius of Circle B: 4 Sum of radii = 2 + 4 = 6 Since √85 is greater than 6, the circles do not overlap. To find the minimum distance between points on both circles, we subtract the sum of the radii from the distance between the centers: Minimum distance = √85 - 6 ≈ 3.82 units.