To find the equation of a circle, we need the center [tex]\((h, k)\)[/tex] and the radius [tex]\(r\)[/tex]. The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((0, 1)\)[/tex].
- The radius [tex]\(r\)[/tex] is 10.
We can substitute the given center and radius into the standard form equation.
1. The center is [tex]\((h, k) = (0, 1)\)[/tex], so [tex]\(h = 0\)[/tex] and [tex]\(k = 1\)[/tex].
2. The radius [tex]\(r\)[/tex] is 10, so [tex]\(r^2 = 10^2\)[/tex]. Calculating the square of the radius, we get:
[tex]\[
r^2 = 100
\][/tex]
Substituting [tex]\(h = 0\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(r^2 = 100\)[/tex] into the standard form equation:
[tex]\[
(x - 0)^2 + (y - 1)^2 = 100
\][/tex]
Simplifying the expression [tex]\((x - 0)^2\)[/tex] to [tex]\(x^2\)[/tex], we get:
[tex]\[
x^2 + (y - 1)^2 = 100
\][/tex]
Therefore, the correct equation of the circle is:
[tex]\[
x^2 + (y - 1)^2 = 100
\][/tex]
So, the correct choice is:
[tex]\[ \boxed{x^2 + (y - 1)^2 = 100} \][/tex]
