Circle A has a center at #(1 ,4 )# and a radius of #2 #. Circle B has a center at #(9 ,3 )# 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 #(1   ,4  )# and a radius of #2 #.  Circle B has a center at #(9   ,3  )# and a radius of #1 #.  Do the circles overlap?  If not what is the smallest distance between them?

Wyatt Anderson · Accepted Answer

#color(green)("They do not overlap.")#
#color(green)("The smallest distance between them is: "sqrt(65)-3"~~5.06# If the distance between centres is less than the sum of the radii then they overlap. Let point 1 be #P_1->(x_1,y_1)=(1,4)# Let point 2 be #P_2->(x_2,y_2)=(9,3)# Let the distance between centres be #d# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine distance between centres")# #x_("difference")=x_2-x_1 = 9-1=8# #y_("difference")=y_2=y_1=3-4=-1# Using Pythagoras #d=sqrt(8^2+(-1)^2) = sqrt(65)# This is approximately #8# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine if they over lap")# The sum of the radii is #2+1=3# So if the sum of the radii is 3 and the distance between centres is very close to 8 #color(green)(ul(" they do not overlap."))# #color(green)("The smallest distance between them is: "sqrt(65)-3"#

Jayden Jackson · Answer

undefined Yes, the circles overlap. The smallest distance between them is 5 units.