Two circles have the following equations #(x +5 )^2+(y +6 )^2= 9 # and #(x +2 )^2+(y -1 )^2= 1 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Compare the distance (d) between the centres of the circles to the sum of the radii.

1) If the sum of the radii #>#d, the circles overlap.
2) If the sum of the radii #<#d, then no overlap.
3) If #d+r_B<= r_A#, then Circle A contains Circle B

Given Circle A, centre #(-5,-6)# and radius #r_A=3#
Circle B, centre #(-2,1),# and radius #r_B=1#

The first step here is to calculate d, use the distance formula :
#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

where #(x_1,y_1) and (x_2,y_2)# are 2 coordinate points

here the two points are #(-5,-6)# and #(-2,1)# the centres of the circles

let#(x_1,y_1)=(-5,-6)# and #(x_2,y_2)=(-2,1)#

#d=sqrt(-2-(-5)^2+(1-(-6)^2)#
#=sqrt(3^2+7^2)=sqrt58=7.6158#

Sum of radii = radius of A #(r_A)#+ radius of B #(r_B)# #= 3+1=4#

Since sum of radius #<#d, then no overlap of the circles
no overlap => no containment

Greatest distance = #d#(the yellow segment) #+r_A+r_B#

#= 7.6158+3+1=11.6158#