Circle A has a center at #(-5 ,8 )# and a radius of #4 #. Circle B has a center at #(-3 ,3 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(-5  ,8   )# and a radius of #4    #.  Circle B has a center at #(-3  ,3   )# and a radius of #4 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Aurora Ayala · Accepted Answer

#"circles overlap"#

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

#"the 2 points here are " (-5,8)" and " (-3,3)#

#"let " (x_1,y_1)=(-5,8)" and " (x_2,y_2)=(-3,3)#

#d=sqrt((-3+5)^2+(3-8)^2)=sqrt(4+25)=sqrt29≈5.385#

#"sum of radii = radius of A + radius of B = 4+4= 8"#

#Since sum of radii > d , then circles overlap. graph{(y^2-16y+x^2+10x+73)(y^2-6y+x^2+6x+2)=0 [-28.48, 28.47, -14.24, 14.24]}

Sadie Allen · Answer

undefined Yes, the circles overlap. The smallest distance between them is 2 units.