To find the equation of a circle centered at the origin with a given radius, we use the standard form of the equation of a circle. The standard form is:
[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.
In this particular problem, the center of the circle is at the origin, [tex]\((0, 0)\)[/tex], and the radius [tex]\(r\)[/tex] is 8. Plugging these values into the standard form, we get:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 8^2
\][/tex]
This simplifies to:
[tex]\[
x^2 + y^2 = 8^2
\][/tex]
Thus, the equation of the circle is:
[tex]\[
x^2 + y^2 = 8^2
\][/tex]
Now, let's match this with the given options:
A. [tex]\((x - 8)^2 + (y - 8)^2 = 64\)[/tex] – This option represents a circle with center [tex]\((8, 8)\)[/tex] and radius [tex]\(\sqrt{64} = 8\)[/tex], which is not centered at the origin, so this is incorrect.
B. [tex]\(x^2 + y^2 = 8^2\)[/tex] – This option exactly matches the equation we derived, so it is correct.
C. [tex]\(x^2 + y^2 = 8\)[/tex] – This option represents a circle with radius [tex]\(\sqrt{8}\)[/tex], which is not the given radius of 8, so this is incorrect.
D. [tex]\(\frac{x^2}{8} + \frac{y^2}{8} = 1\)[/tex] – This option represents an ellipse, not a circle, so it is incorrect.
Therefore, the correct option is:
[tex]\[ \boxed{B} \][/tex]
