From the circle A: (x-5) 2+(y+7) 2 = 16, it is concluded that the coordinate of A is (5, -7), the radius R=4, and the radius R = 1 of circle B.

According to the image:

When circle B is inscribed with circle A, the locus of center B is a circle with center A and radius equal to R-r=4- 1=3.

Then the equation of circle B is: (x-5) 2+(y+7) 2 = 9;

When circle B is circumscribed with circle A, the locus of center B is a circle with center A and radius equal to R+r=4+ 1=5.

Then the equation of circle B is: (x-5) 2+(y+7) 2 = 25.

To sum up, the trajectory equation of the center of motion is: (x-5)2+(y+7)2=25 or (x-5) 2+(y+7) 2 = 9.