To determine which equation represents a circle with a center at [tex]\( T(5, -1) \)[/tex] and a radius of 16 units, we use the standard form of the equation of a circle. The standard form 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]\( T(5, -1) \)[/tex]. Thus, [tex]\( h = 5 \)[/tex] and [tex]\( k = -1 \)[/tex].
- The radius of the circle is [tex]\( 16 \)[/tex] units, so [tex]\( r = 16 \)[/tex].
First, square the radius to find [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = 16^2 = 256
\][/tex]
Now, substitute [tex]\( h = 5 \)[/tex], [tex]\( k = -1 \)[/tex], and [tex]\( r^2 = 256 \)[/tex] into the standard form equation:
[tex]\[
(x - 5)^2 + (y + 1)^2 = 256
\][/tex]
This is the equation representing the circle with the given center and radius. Comparing this to the given options:
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]
We see that option B matches our derived equation:
[tex]\((x - 5)^2 + (y + 1)^2 = 256\)[/tex]
Thus, the correct answer is:
[tex]\[
\boxed{B}
\][/tex]
