Circle A has a radius of #2 # and a center at #(5 ,1 )#. Circle B has a radius of #1 # and a center at #(3 ,4 )#. If circle B is translated by #<-2 ,6 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
May 10, 2016

no overlap , ≈ 6.849

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

Firstly ,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 of #((-2),(6))#

centre B(3 ,4) → (3-2 ,4+6) → (1 ,10)

To calculate d , 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)=(5,1)" and (x_2,y_2)=(1,10)#

#d=sqrt((1-5)^2+(10-1)^2)=sqrt(16+81)=sqrt97 ≈ 9.849#

radius of A + radius of B = 2 + 1 = 3

Since sum of radii < d , then no overlap

min. distance between points = 9.849 - 3 = 6.849
graph{(y^2-2y+x^2-10x+22)(y^2-20y+x^2-2x+100)=0 [-40, 40, -20, 20]}