Circle A has a center at #(9 ,4 )# and a radius of #4 #. Circle B has a center at #(-2 ,6 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(9  ,4   )# and a radius of #4    #.  Circle B has a center at #(-2   ,6   )# and a radius of #5 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Avery Allen · Accepted Answer

no overlap, smallest distance ≈ 2.18

What we have to do here is #color(blue)"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)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))# where # (x_1,y_1),(x_2,y_2)" are 2 points"# The 2 points here are (9 ,4) and (-2 ,6) let # (x_1,y_1)=(9,4)" and " (x_2,y_2)=(-2,6)# #d=sqrt((-2-9)^2+(6-4)^2)=sqrt(121+4)≈11.18# Sum of radii = radius of A + radius of B = 4 + 5 = 9 Since sum of radii < d , then no overlap smallest distance between them = d - sum of radii #=11.18-9=2.18# graph{(y^2-8y+x^2-18x+81)(y^2-12y+x^2+4x+15)=0 [-22.8, 22.8, -11.4, 11.41]}

Aurora Ayala · Answer

undefined No, the circles do not overlap. The smallest distance between them is 6.