To determine the radius and the center of the circle given by the equation [tex]\(x^2 + (y - 10)^2 = 16\)[/tex], let's start by analyzing the equation in its general form.
The general form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] represents the radius of the circle.
Now, let's match the given equation [tex]\(x^2 + (y - 10)^2 = 16\)[/tex] to this general form:
1. Compare the given equation with the general form:
- We see that [tex]\(x^2\)[/tex] can be written as [tex]\((x - 0)^2\)[/tex]. This implies that there is no horizontal shift, so the x-coordinate of the center is [tex]\(h = 0\)[/tex].
- [tex]\((y - 10)^2\)[/tex] indicates that the circle is shifted 10 units upwards in the y-direction, so the y-coordinate of the center is [tex]\(k = 10\)[/tex].
Therefore, the center [tex]\((h, k)\)[/tex] of the circle is:
[tex]\[
(0, 10)
\][/tex]
2. Next, determine the radius [tex]\(r\)[/tex] of the circle. The right-hand side of the equation [tex]\(16\)[/tex] represents [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = 16
\][/tex]
To find the radius [tex]\(r\)[/tex], take the square root of both sides:
[tex]\[
r = \sqrt{16} = 4
\][/tex]
Therefore, the radius of the circle is:
[tex]\[
4 \text{ units} \checkmark
\][/tex]
In conclusion:
- The center of the circle is at [tex]\((0, 10)\)[/tex].
- The radius of the circle is [tex]\(4 \text{ units} \checkmark\)[/tex].
