# Circle A has a radius of #2 # and a center at #(5 ,6 )#. Circle B has a radius of #5 # and a center at #(2 ,4 )#. If circle B is translated by #<-2 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

##### 1 Answer

circles overlap

#### Explanation:

What we have to do here is to compare the distance (d) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

However we require to find the new centre of B under the translation.

A translation does not change the shape of a figure only it's position.Under a translation

#((-2),(1))# centre B(2 ,4) → (-2+2 ,4+1) → (0 ,5)

To calculate the distance (d) between centres use the

#color(blue)" distance formula "#

#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|))#

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

# (x_1,y_1)=(5,6)" and " (x_2,y_2)=(0,5)#

#rArr d=sqrt((0-5)^2+(5-6)^2)=sqrt26 ≈ 5.099# radius of A + radius of B = 2 + 5 = 7

Since sum of radii > d , then circles overlap

graph{(y^2-12y+x^2-10x+57)(y^2-10y+x^2)=0 [-20, 20, -10, 10]}