To determine the equation of a circle, we use the standard form of the equation of a circle, which 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.
Given:
- The center of the circle is [tex]\((8, 10)\)[/tex], so [tex]\(h = 8\)[/tex] and [tex]\(k = 10\)[/tex].
- The radius of the circle is [tex]\(6\)[/tex], so [tex]\(r = 6\)[/tex].
Now, let's substitute these values into the standard form equation for a circle.
1. Substitute [tex]\(h = 8\)[/tex], [tex]\(k = 10\)[/tex], and [tex]\(r = 6\)[/tex] into the equation:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 6^2
\][/tex]
2. Simplify the radius squared:
[tex]\[
6^2 = 36
\][/tex]
3. Therefore, the equation becomes:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 36
\][/tex]
So, the equation of the circle is:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 36
\][/tex]
From the given options, this matches option 3:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 36
\][/tex]
Thus, the correct choice is:
Choice 3.
