Answer: [tex](\text{x}-1)^2+(\text{y}+6)^2 = 9[/tex]
Explanation
The highest and lowest points on the circle are at (1,-3) and (1,-9) respectively.
Apply the midpoint formula on them to determine the center is at (1,-6). Effectively you are just finding the midpoint of y = -3 and y = -9 to get y = (-3 + -9)/2 = -12/2 = -6. The x coordinate is stuck at x = 1 the entire time.
We determine that h = 1 and k = -6 because the center is located at (h,k)
The distance from (1,-3) to (1,-9) is 6 units. This is the diameter of the circle. Half of which is the radius r = 3.
Plug h = 1, k = -6, r = 3 into the template below to finish up the problem.
[tex](\text{x}-h)^2+(\text{y}-k)^2 = r^2\\\\(\text{x}-1)^2+(\text{y}-(-6))^2 = 3^2\\\\(\text{x}-1)^2+(\text{y}+6)^2 = 9\\\\[/tex]
I used the graphing tool GeoGebra to confirm the answer is correct. Desmos can be used as well.