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This circle is centered at the origin, and the length of its radius is 10. What is the circle's equation?

A. [tex] x^{10} + y^{10} = 100 [/tex]

B. [tex] x + y = 10 [/tex]

C. [tex] x^2 + y^2 = 10 [/tex]

D. [tex] x^2 + y^2 = 100 [/tex]


Sagot :

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]