To find the equation of a circle centered at the origin with a given radius, we use the standard form of the circle equation:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
In this problem, the radius [tex]\( r \)[/tex] is given as 3. Therefore, to find the equation, we need to substitute [tex]\( r \)[/tex] with 3 in the standard form.
First, calculate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = 3^2 = 9 \][/tex]
Now, replace [tex]\( r^2 \)[/tex] in the standard form:
[tex]\[ x^2 + y^2 = 9 \][/tex]
So the equation of the circle is:
[tex]\[ x^2 + y^2 = 9 \][/tex]
Now, let's look at the provided options and see which one matches our equation:
A. [tex]\( x^2 + y^2 = 3 \)[/tex]
B. [tex]\( x^3 + y^3 = 27 \)[/tex]
C. [tex]\( x^2 + y^2 = 9 \)[/tex]
D. [tex]\( x + y = 3 \)[/tex]
Clearly, option C ([tex]\( x^2 + y^2 = 9 \)[/tex]) matches the equation we derived. Therefore, the correct answer is:
C. [tex]\( x^2 + y^2 = 9 \)[/tex]
