To solve the problem, we'll start by comparing the given equation of the circle to the standard form of the equation of a circle.
The standard form of the equation of a circle is:
[tex]$(x - h)^2 + (y - k)^2 = r^2$[/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] represents the radius.
Given the equation:
[tex]$x^2 + (y - 10)^2 = 16$[/tex]
### Step 1: Identify the center of the circle
By comparing the given equation to the standard form, we can see that [tex]\(x^2\)[/tex] can be written as [tex]\((x - 0)^2\)[/tex]. This means the center in the x-direction is at [tex]\(h = 0\)[/tex].
Similarly, [tex]\((y - 10)^2\)[/tex] indicates that the center in the y-direction is at [tex]\(k = 10\)[/tex].
Therefore, the center of the circle is at:
[tex]\[(0, 10)\][/tex]
### Step 2: Identify the radius of the circle
In the standard equation [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], the term on the right-hand side represents [tex]\(r^2\)[/tex]. From the given equation [tex]\(x^2 + (y - 10)^2 = 16\)[/tex], we can see that:
[tex]\[r^2 = 16\][/tex]
To find the radius [tex]\(r\)[/tex], we take the square root of both sides:
[tex]\[r = \sqrt{16}\][/tex]
[tex]\[r = 4\][/tex]
### Final Answer
The radius of the circle is [tex]\(4\)[/tex] units.
The center of the circle is at [tex]\((0, 10)\)[/tex].
