Circle A has a radius of #3 # and a center of #(2 ,1 )#. 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
Jul 11, 2016

circles overlap

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

The first step is to find the coordinates of B under the translation. A translation does not change the shape of the circle only it's position.

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

B(7 ,3) → (7-3 ,3+2) → B(4 ,5) is new position of centre B

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 coordinate points"#

Here the 2 points are A(2 ,1) and B(4 ,5)

let # (x_1,y_1)=(2,1)" and " (x_2,y_2)=(4,5)#

#d=sqrt((4-2)^2+(5-1)^2)=sqrt(4+16)=sqrt20≈4.472#

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

Since sum of radii > d , then circles overlap
graph{(y^2-2y+x^2-4x-5)(y^2-10y+x^2-8x+25)=0 [-20, 20, -10, 10]}