Find the equations for two different circles with radius 4 which pass through the point (1, 1)

standard form of a circle is (x-h)²+(y-k)²=r², (h, k) being the center, r the radius.
In this case, (x-h)²+(y-k)²=16
the circles pass through (1,1), that is x=1, y=1
Plug x=1,y=1 in the equation: (1-h)²+(1-k)²=16
if h=1, k is either -3 or 5
so the equation is (x-1)²+(y+3)²=16 or (x-1)²+(y-5)²=16

You can also set k=1, h=-3 or 5
Note: there are many other possibilities.