The question asks for the center-radius form of the equation of a circle with a center at $(4, 8)$ and a radius of 8.

The general formula for the equation of a circle is:

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

where:

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

Given:

• Center: $(4, 8)$, so $h = 4$ and $k = 8$,
• Radius: $r = 8$, so $r^2 = 64$.

Substitute these values into the formula:

$(x - 4)^2 + (y - 8)^2 = 64$

This matches option A in the multiple-choice list.

So, the correct answer is A: $(x - 4)^2 + (y - 8)^2 = 64$.

Would you like more details or have any questions about this? Here are five related questions to deepen your understanding:

1. What would the equation look like if the radius were doubled?
2. How would the equation change if the center were moved to $(-4, -8)$?
3. How do you find the equation of a circle given a point on its circumference and the center?
4. What is the geometric significance of the terms $(x - h)^2$ and $(y - k)^2$?
5. Can you find the equation of a circle passing through the origin with a center at $(4, 8)$?

Tip: Remember, the sign inside the parentheses in the equation of a circle is always opposite to the coordinates of the center!