To determine which equation correctly represents a circle centered at the origin with a radius of 10, we need to recall the standard form of the equation for a circle. The standard form is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
For a circle centered at the origin [tex]\((0,0)\)[/tex] with a radius of 10, we substitute [tex]\(h = 0\)[/tex], [tex]\(k = 0\)[/tex], and [tex]\(r = 10\)[/tex] into the standard form. This gives us the equation:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 10^2
\][/tex]
Simplifying this, we have:
[tex]\[
x^2 + y^2 = 10^2
\][/tex]
And since [tex]\(10^2\)[/tex] equals 100, this further simplifies to:
[tex]\[
x^2 + y^2 = 100
\][/tex]
So the equation that represents a circle centered at the origin with a radius of 10 is:
[tex]\[
x^2 + y^2 = 100
\][/tex]
Among the given options, the correct one is:
B. [tex]\(x^2 + y^2 = 100\)[/tex]
