The center of circle #(x+2)^2+(y-5)^2=16# is #(-2,5)# and radius is #4# and center of circle #(x+4)^2+(y-3)^2=49# is #(-4,3)# and radius is #7#.
The distance between centers is #sqrt(((-4)-(-2))^2+(3-5)^2#
= #sqrt(4+4)=sqrt8=2sqrt2=2.8284#
If the radii of two circles is #r_1# and #r_2# and we also assume that #r_1>r_2# and the distance between centers is #d#, then
A - if #r_1+r_2=d#, they touch each other externally and greatest possible distance between a point on one circle and another point on the other is #2(r_1+r_2)# and smallest possible distance between a point on one circle and another point on the other is #0#.
B - if #r_1+r_2 < d#, they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest possible distance between a point on one circle and another point on the other is #d-r_1-r_2#.
C - if #r_1+r_2 > d# and #r_1-r_2=d#, they touch each other internally and smaller circle is contained in other and greatest possible distance between a point on one circle and another point on the other is #2r_2# and smallest distance is #0#.
D - if #r_1+r_2 > d# and #r_1-r_2>d#, smaller circle lies inside the larger circle and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest distance is #r_1-r_2-d#.
E - if #r_1+r_2 > d# and #r_1-r_2 < d#, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest distance is #0#.
Now, here as #r_1+r_2=11>d=2.8284# and #r_1-r_2=3>d=2.8284#. Hence, smaller circle lies inside the larger circle and hence is contained in bigger circle. The greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d=7+4+2.8284=13.8284#.
graph{(x^2+y^2+4x-10y+13)(x^2+y^2+8x-6y-24)=0 [-22, 18, -5.36, 14.64]}
