To find the equation of a circle, we use the standard form of a circle's equation:

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

Where:

• $(h, k)$ is the center of the circle.
• $r$ is the radius of the circle.

Given Information:

• The center is $(0, -4)$.
• The circle passes through the point $(-4, 2)$.

Step 1: Substituting the center $(0, -4)$ into the equation.

The equation becomes: $(x - 0)^2 + (y + 4)^2 = r^2$ Simplified: $x^2 + (y + 4)^2 = r^2$

Step 2: Finding the radius $r$.

We use the fact that the point $(-4, 2)$ lies on the circle to find the radius.

Substitute $(-4, 2)$ into the equation: $(-4)^2 + (2 + 4)^2 = r^2$ $16 + 6^2 = r^2$ $16 + 36 = r^2$ $r^2 = 52$

Step 3: Final Equation of the Circle

Now that we have $r^2 = 52$, the equation of the circle is: $x^2 + (y + 4)^2 = 52$

This is the equation of the circle.

Would you like more details or further clarifications? Here are some related questions for further exploration:

1. How do you derive the general equation of a circle from three points?
2. Can a circle have a negative radius? Why or why not?
3. What happens when the center of a circle is shifted along the axes?
4. How would the equation change if the circle was tangent to the x-axis?
5. How do you determine if a point lies inside, on, or outside the circle?

Tip: The distance formula is key when solving problems involving radii or distances between points on a plane!