Let's solve the problem step by step to find the correct equation of the circle given its center and radius.
1. Identify the given information:
- The center of the circle is [tex]\( T(5, -1) \)[/tex].
- The radius of the circle is 16 units.
2. Recall the standard form of the equation of a circle:
The standard form of a circle's equation with center [tex]\((h,k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
3. Substitute the given center and radius into the standard equation form:
- Given center [tex]\((h,k) = (5, -1)\)[/tex],
- Given radius [tex]\(r = 16\)[/tex],
Substitute [tex]\(h = 5\)[/tex], [tex]\(k = -1\)[/tex], and [tex]\(r = 16\)[/tex] into the standard equation:
[tex]\[
(x - 5)^2 + (y - (-1))^2 = 16^2
\][/tex]
4. Simplify the equation:
- [tex]\(y - (-1)\)[/tex] simplifies to [tex]\(y + 1\)[/tex],
- [tex]\(16^2\)[/tex] calculates to 256.
Therefore, the equation now looks like:
[tex]\[
(x - 5)^2 + (y + 1)^2 = 256
\][/tex]
5. Verify which of the given multiple-choice options matches our derived equation:
- Option A: [tex]\((x - 5)^2 + (y + 1)^2 = 16\)[/tex] [tex]\( \quad \)[/tex] (Incorrect, as the right-hand side should be 256, not 16)
- Option B: [tex]\((x - 5)^2 + (y + 1)^2 = 256\)[/tex] [tex]\( \quad \)[/tex] (Correct, matches our derived equation)
- Option C: [tex]\((x + 5)^2 + (y - 1)^2 = 16\)[/tex] [tex]\( \quad \)[/tex] (Incorrect, as both the signs and the right-hand side values are incorrect)
- Option D: [tex]\((x + 5)^2 + (y - 1)^2 = 256\)[/tex] [tex]\( \quad \)[/tex] (Incorrect, the signs inside the parentheses are incorrect)
Therefore, the correct choice is:
[tex]\[
\boxed{B}
\][/tex]
