Circle A has a center at #(3 ,-1 )# and a radius of #6 #. Circle B has a center at #(-2 ,-4 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?

1 Answer
Sep 22, 2016

circles overlap.

Explanation:

What we have to do here is compare the distance ( d) between the centres of the circles to the #color(blue)"sum of the radii"#

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

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"#

The 2 points here are (3 ,-1) and (-2 ,-4)

let # (x_1,y_1)=(3,-1)" and " (x_2,y_2)=(-2,-4)#

#d=sqrt((-2-3)^2+(-4+1)^2)=sqrt(25+9)≈5.831#

sum of radii = radius of A + radius of B = 6 + 3 = 9

Since sum of radii > d , then the circles overlap
graph{(y^2+2y+x^2-6x-26)(y^2+8y+x^2+4x+11)=0 [-20, 20, -10, 10]}