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 :

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]