To solve for the radius and the center of the circle given the equation [tex]\(x^2 + (y - 10)^2 = 16\)[/tex], we can follow these steps:
1. Identify the Standard Form: The standard form of the equation of a circle is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle, and [tex]\(r\)[/tex] represents the radius of the circle.
2. Compare with the Given Equation: The given equation is [tex]\(x^2 + (y - 10)^2 = 16\)[/tex]. We need to compare this with the standard form to identify [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r^2\)[/tex].
3. Determine the Center:
- Notice that the term [tex]\((x - h)^2\)[/tex] is [tex]\((x - 0)^2\)[/tex] because it is simply [tex]\(x^2\)[/tex]. This means [tex]\(h = 0\)[/tex].
- Also, the term [tex]\((y - k)^2\)[/tex] is given as [tex]\((y - 10)^2\)[/tex]. This means [tex]\(k = 10\)[/tex].
- Therefore, the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((0, 10)\)[/tex].
4. Determine the Radius:
- The right side of the equation is [tex]\(16\)[/tex], which corresponds to [tex]\(r^2\)[/tex] in the standard form. Therefore, [tex]\(r^2 = 16\)[/tex].
- To find the radius [tex]\(r\)[/tex], we take the square root of [tex]\(16\)[/tex], giving us [tex]\(r = 4\)[/tex].
So, the radius of the circle is [tex]\(\boxed{4}\)[/tex] units, and the center of the circle is at [tex]\(\boxed{(0, 10)}\)[/tex].
