The problem asks us to find the equation of a circle given its center and diameter. The center is at $(-2, 6)$ and the diameter is $8$.

1. Problem Description

The problem asks us to find the equation of a circle given its center and diameter. The center is at $(-2, 6)$ and the diameter is $8$.

2. Solution Steps

The general equation of a circle with center $(h, k)$ and radius $r$ is:

$(x - h)^2 + (y - k)^2 = r^2$

In this problem, we are given the center $(h, k) = (-2, 6)$ and the diameter $d = 8$. The radius is half of the diameter, so $r = d/2 = 8/2 = 4$.

Therefore, $r^2 = 4^2 = 16$.

Substituting the values of $h$, $k$, and $r^2$ into the general equation of a circle, we have:

$(x - (-2))^2 + (y - 6)^2 = 16$

$(x + 2)^2 + (y - 6)^2 = 16$

3. Final Answer

The equation of the circle is $(x + 2)^2 + (y - 6)^2 = 16$. Therefore, the correct answer is C.