Circle A has a center at #(5 ,-2 )# and a radius of #2 #. Circle B has a center at #(1 ,-4 )# and a radius of #1 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(5  ,-2   )# and a radius of #2    #.  Circle B has a center at #(1   ,-4   )# and a radius of #1 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Sadie Allen · Accepted Answer

no overlap, smallest distance ≈ 1.47 units

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 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 coordinate points"# The 2 points here are (5 ,-2) and (1 ,-4) let # (x_1,y_1)=(5,-2)" and " (x_2,y_2)=(1,-4)# #d=sqrt((1-5)^2+(-4+2)^2)=sqrt(16+4)=sqrt20≈4.47# sum of radii = radius of A + radius of B = 2 + 1 = 3 Since sum of radii < d, then no overlap of circles smallest distance between circles = d - sum of radii #=4.47-3=1.47# graph{(y^2+4y+x^2-10x+25)(y^2+8y+x^2-2x+16)=0 [-10, 10, -5, 5]}