To determine which of the given equations correctly represents a circle centered at the origin with a radius of 10, we need to recall the standard equation for a circle.
The standard form of the equation of a circle centered at the origin [tex]\((0,0)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
In this case, the radius [tex]\(r\)[/tex] is given as 10. Therefore, substituting [tex]\(r = 10\)[/tex] into the standard form, we get:
[tex]\[ x^2 + y^2 = 10^2 \][/tex]
Simplifying [tex]\(10^2\)[/tex], we get:
[tex]\[ x^2 + y^2 = 100 \][/tex]
Now, let's examine each of the given answer choices:
A. [tex]\( x^2 + y^2 = 100^2 \)[/tex]
- This simplifies to [tex]\( x^2 + y^2 = 10000 \)[/tex], which represents a circle with radius 100, not 10. Hence, this is incorrect.
B. [tex]\( x^2 + y^2 = 100 \)[/tex]
- This matches our derived equation [tex]\( x^2 + y^2 = 100 \)[/tex], which represents a circle with radius 10 centered at the origin. Hence, this is correct.
C. [tex]\( x^2 + y^2 = 10 \)[/tex]
- This represents a circle with radius [tex]\( \sqrt{10} \)[/tex], which is not 10. Hence, this is incorrect.
D. [tex]\( (x-10)^2 + (y-10)^2 = 100 \)[/tex]
- This equation represents a circle centered at (10, 10) with radius 10. While it has the correct radius, the center is not at the origin. Hence, this is incorrect.
From these checks, we can conclude that the correct equation is:
[tex]\[ \text{B. } x^2 + y^2 = 100 \][/tex]
Thus, the correct index for the equation representing a circle centered at the origin with a radius of 10 is:
[tex]\[ \boxed{2} \][/tex]
And the radius squared is:
[tex]\[ 100 \][/tex]
So, the final correct information is:
[tex]\[ (100, 2) \][/tex]
In summary, the equation [tex]\( x^2 + y^2 = 100 \)[/tex] correctly represents a circle centered at the origin with a radius of 10.
