To find the equation of the circle that is centered at the origin with a radius of 4, we start by recalling the general form of the equation of a circle centered at the origin. The equation is given by:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
Given that the radius [tex]\( r \)[/tex] is 4, we will substitute [tex]\( r = 4 \)[/tex] into the equation. This gives us:
[tex]\[ x^2 + y^2 = 4^2 \][/tex]
We can further simplify [tex]\( 4^2 \)[/tex] as 16, giving us:
[tex]\[ x^2 + y^2 = 16 \][/tex]
Now, let's evaluate the given options:
- A. [tex]\((x-4)^2 + (y-4)^2 = 16\)[/tex] is incorrect because it represents a circle centered at (4,4), not the origin.
- B. [tex]\(x^2 + y^2 = 4\)[/tex] is incorrect because this represents a circle with a radius of 2, not 4.
- C. [tex]\(\frac{x^2}{4} + \frac{y^2}{4} = 1\)[/tex] is incorrect because this is the equation of an ellipse.
- D. [tex]\(x^2 + y^2 = 4^2\)[/tex] is the correct option because it can be simplified to [tex]\(x^2 + y^2 = 16\)[/tex], which fits our derived equation.
Therefore, the correct equation of the circle is:
[tex]\[ \boxed{x^2 + y^2 = 4^2} \][/tex]
So the answer is:
[tex]\[ \boxed{4} \][/tex]
