Answer: B. $(x-4)^{2}+(y-2)^{2}=36$

Explanation:1. Recall the standard equation of a circle: The equation of a circle with center $(h,k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.

2. Determine the center and radius of Circle A: The equation of Circle A is $(x-4)^2 + (y-2)^2 = 9$. From this equation: The center of Circle A is $(h,k) = (4,2)$. The radius squared of Circle A is $r\_A^2 = 9$. So, the radius of Circle A is $r\_A = \sqrt{9} = 3$.

3. Determine the center and radius of Circle B: Circle B has the same center as Circle A. So, the center of Circle B is also $(h,k) = (4,2)$. The radius of Circle B ($r\_B$) is twice the radius of Circle A. $r\_B = 2 \times r\_A = 2 \times 3 = 6$.

4. Calculate the radius squared for Circle B: $r\_B^2 = (6)^2 = 36$.

5. Write the equation for Circle B: Using the standard form $(x-h)^2 + (y-k)^2 = r\_B^2$, with center $(4,2)$ and $r\_B^2 = 36$: The equation of Circle B is $(x-4)^2 + (y-2)^2 = 36$.

6. Check the solution with the options: The equation for Circle B is $(x-4)^{2}+(y-2)^{2}=36$. This matches option [B].

7. Conclusion: The equation that represents circle B is $(x-4)^{2}+(y-2)^{2}=36$.