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?

1 Answer
Aug 7, 2016

#color(green)("They do not overlap.")#
#color(green)("The smallest distance between them is: "sqrt(65)-3"~~5.06#

Explanation:

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"#