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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!

Find the center-radius form of the equation of the circle with center (4,8) and radius 8.

Solve the equation of a circle with center (4,8) and radius 8. This problem involves geometry and the standard form equation of a circle, suitable for high school students in Grades 9-12.

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