To determine the correct equation that represents a circle centered at the origin with a radius of 10, we need to use the standard equation of a circle.
The general form for the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Since the circle is centered at the origin [tex]\((0, 0)\)[/tex] and has a radius of 10, we substitute [tex]\(h = 0\)[/tex], [tex]\(k = 0\)[/tex], and [tex]\(r = 10\)[/tex]:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 10^2
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 + y^2 = 100
\][/tex]
Among the given options:
A. [tex]\((x-10)^2 + (y-10)^2 = 100\)[/tex] represents a circle with center [tex]\((10, 10)\)[/tex] and radius [tex]\(\sqrt{100} = 10\)[/tex], which is incorrect for a circle centered at the origin.
B. [tex]\(x^2 + y^2 = 100\)[/tex] correctly represents a circle centered at the origin with radius 10.
C. [tex]\(x^2 + y^2 = 10\)[/tex] represents a circle with radius [tex]\(\sqrt{10}\)[/tex], which is incorrect.
D. [tex]\(x^2 + y^2 = 100^2\)[/tex] represents a circle with radius 100, which is incorrect.
Therefore, the correct answer is:
Option B: [tex]\(x^2 + y^2 = 100\)[/tex]
