## Answer :

[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.

Given that the circle is centered at the origin [tex]\((0, 0)\)[/tex] and has a radius of 8, we can substitute these values into the standard form equation:

- Center [tex]\((h, k) = (0, 0)\)[/tex]

- Radius [tex]\(r = 8\)[/tex]

Substituting these values, the equation becomes:

[tex]\[ (x - 0)^2 + (y - 0)^2 = 8^2 \][/tex]

Simplifying this, we get:

[tex]\[ x^2 + y^2 = 64 \][/tex]

Now, let's examine the given options and see which one matches this equation:

A. [tex]\( x^2 + y^2 = 8^2 \)[/tex]

B. [tex]\( x^2 + y^2 = 8 \)[/tex]

C. [tex]\( \frac{x^2}{8} + \frac{y^2}{8} = 1 \)[/tex]

D. [tex]\( (x - 8)^2 + (y - 8)^2 = 64 \)[/tex]

- Option A: [tex]\( x^2 + y^2 = 8^2 \)[/tex] simplifies to [tex]\( x^2 + y^2 = 64 \)[/tex], which is exactly our equation.

- Option B: [tex]\( x^2 + y^2 = 8 \)[/tex] is incorrect because it does not correctly represent the radius of 8.

- Option C: [tex]\( \frac{x^2}{8} + \frac{y^2}{8} = 1 \)[/tex] is incorrect as it resembles the equation of an ellipse, not a circle.

- Option D: [tex]\( (x - 8)^2 + (y - 8)^2 = 64 \)[/tex] correctly represents a circle, but centered at (8, 8), not the origin.

Therefore, the correct answer is:

[tex]\[ \boxed{A. \, x^2 + y^2 = 8^2} \][/tex]