Circle A has a radius of #3 # and a center of #(2 ,6 )#. Circle B has a radius of #4 # and a center of #(7 ,3 )#. If circle B is translated by #<-3 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
May 12, 2016

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 the sum of radii > d , then circles overlap

• If the sum of radii < d , then no overlap

Firstly we require to find the coordinates of the centre of B under the given translation.

Under a translation of #((-3),(2))#

centre B (7 ,3) → (7-3 ,3+2) → (4 ,5)

To calculate the distance (d) between the 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)" and " (x_2,y_2)" are 2 coord points"#

let # (x_1,y_1)=(2,6)" and " (x_2,y_2)=(4,5)#

#d=sqrt((4-2)^2+(5-6)^2)=sqrt(4+1)=sqrt5 ≈ 2.236#

radius of A + radius of B = 3 + 4 = 7

Since sum of radii > d , then circles overlap
graph{(y^2-12y+x^2-4x+31)(y^2-10y+x^2-8x+25)=0 [-15.59, 15.59, -7.8, 7.79]}