To determine the equation of a circle centered at the origin (0,0) with a radius of 10, we need to follow these steps:
1. Recall the general form of the equation of a circle:
The equation of a circle centered at the origin [tex]\((0,0)\)[/tex] with radius [tex]\(r\)[/tex] is given by:
[tex]\[
x^2 + y^2 = r^2
\][/tex]
2. Identify the given radius:
We are given that the radius [tex]\(r\)[/tex] is 10.
3. Substitute the radius into the equation:
We substitute [tex]\(r = 10\)[/tex] into the general equation.
[tex]\[
x^2 + y^2 = 10^2
\][/tex]
4. Simplify the equation:
Calculate the square of the radius.
[tex]\[
10^2 = 100
\][/tex]
So, the equation becomes:
[tex]\[
x^2 + y^2 = 100
\][/tex]
Now, let's compare this with the given options:
- Option A: [tex]\(x^{10}+y^{10}=100\)[/tex] – This is incorrect because the exponents should be 2, not 10.
- Option B: [tex]\(x+y=10\)[/tex] – This represents a linear equation, not a circle.
- Option C: [tex]\(x^2+y^2=10\)[/tex] – This represents a circle, but with the wrong radius squared.
- Option D: [tex]\(x^2+y^2=100\)[/tex] – This matches our calculated equation perfectly.
Therefore, the correct option is:
[tex]\[
\boxed{x^2 + y^2 = 100}
\][/tex]
