1. **State the problem:** We need to find the equation of a circle given its center and radius. 2. **Recall the formula:** The standard form of a circle's equation with center $(h, k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Identify the given values:** The center is $(-4, 6)$ and the radius is $2$. 4. **Substitute the values into the formula:** $$ (x - (-4))^2 + (y - 6)^2 = 2^2 $$ which simplifies to $$ (x + 4)^2 + (y - 6)^2 = 4 $$ 5. **Final answer:** The equation of the circle is $$ (x + 4)^2 + (y - 6)^2 = 4 $$ This equation represents all points $(x, y)$ that are exactly 2 units away from the center $(-4, 6)$ on the coordinate plane.