To find the correct equation of a circle that is centered at the origin with a given radius, we can use the standard form of the equation of a circle:
[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]
