The center of circle (x-4)^2+(y+3)^2=9 is (2,-3) and radius is 3 and center of circle (x+4)^2+(y-1)^2=16 is (-4,1) and radius is 4.
The distance between centers is sqrt((-4-2)^2+(1-(-3))^2
= sqrt(36+16)=sqrt52=7.2111
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=7 < d=7.2111 and 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 4+3+7.2111=14.2111.
graph{(x^2+y^2-8x+6y+16)(x^2+y^2+8x-2y+1)=0 [-10.5, 9.5, -5.665, 4.755]}
