Two circles have the following equations #(x +5 )^2+(y +6 )^2= 9 # and #(x +2 )^2+(y -1 )^2= 81 #. 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?

The given equations of the circle are
#(x+5)^2+(y+6)^2=9" "#first circle
#(x+2)^2+(y-1)^2=81" "#second circle

We start with the equation passing thru the centers of the circle

#C_1(x_1, y_1)=(-5, -6)# and #C_2(x_2, y_2)=(-2, 1)# are the centers.

Using two-point form

#y-y_1=((y_2-y_1)/(x_2-x_1))*(x-x_1)#

#y--6=((1--6)/(-2--5))*(x--5)#

#y+6=((1+6)/(-2+5))*(x+5)#

#y+6=((7)/(3))*(x+5)#

After simplification

#3y+18=7x+35#

#7x-3y=-17" "#equation of the line passing thru the centers and at the two points farthest to each other.

Solve for the points using first circle and the line
#(x+5)^2+(y+6)^2=9" "#first circle
#7x-3y=-17" "#the line

One point at #A(x_a, y_a)=(-6.1817578957376, -8.7574350900543)#
Another at #B(x_b, y_b)=(-3.8182421042626, -3.2425649099459)#
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Solve for the points using second circle and the line
#(x+2)^2+(y-1)^2=81" "#second circle
#7x-3y=-17" "#the line

One point at #C(x_c, y_c)=(1.5452736872127, 9.2723052701629)#
Another at #D(x_d, y_d)=(-5.5452736872127, -7.2723052701625)#
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To compute for the farthest distance #d_f# we will use point #A# and #C#

#d_f=sqrt((x_a-x_c)^2+(y_a-y_c)^2)#

#d_f=sqrt((-6.1817578957376-1.5452736872127)^2+(-8.7574350900543-9.2723052701629)^2)#

#color(blue)(d_f=19.615773105864" "#unit)s

Kindly see the graph

God bless .... I hope the explanation is useful.