To determine the equation of a circle with a given center and radius, we can use the standard form of the circle's equation:
[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 [tex]\(T(5, -1)\)[/tex] means [tex]\(h = 5\)[/tex] and [tex]\(k = -1\)[/tex].
- Radius [tex]\(r = 16\)[/tex].
Now, substitute the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] into the standard form equation:
[tex]\[
(x - 5)^2 + (y + 1)^2 = 16^2
\][/tex]
Next, calculate [tex]\(16^2\)[/tex]:
[tex]\[
16^2 = 256
\][/tex]
So the equation becomes:
[tex]\[
(x - 5)^2 + (y + 1)^2 = 256
\][/tex]
Given the options, we can see that this matches option B:
B. [tex]\((x - 5)^2 + (y + 1)^2 = 256\)[/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B}
\][/tex]
