Select the correct answer.

Which equation represents a circle with center [tex]\( T(5, -1) \)[/tex] and a radius of 16 units?

A. [tex]\((x-5)^2 + (y+1)^2 = 16\)[/tex]

B. [tex]\((x-5)^2 + (y+1)^2 = 256\)[/tex]

C. [tex]\((x+5)^2 + (y-1)^2 = 16\)[/tex]

D. [tex]\((x+5)^2 + (y-1)^2 = 256\)[/tex]



Answer :

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]