To find the coordinates of the center and the radius of the circle given by the equation [tex]\( x^2 + y^2 - x - 2y - \frac{11}{4} = 0 \)[/tex], we need to rewrite the equation in the standard form of a circle, which is [tex]\( (x-h)^2 + (y-k)^2 = r^2 \)[/tex]. Here, [tex]\((h, k)\)[/tex] will be the coordinates of the center, and [tex]\( r \)[/tex] will be the radius.
### Step-by-Step Solution
1. Rewrite the given equation:
[tex]\[
x^2 + y^2 - x - 2y - \frac{11}{4} = 0
\][/tex]
2. Group the [tex]\( x \)[/tex] terms and [tex]\( y \)[/tex] terms:
[tex]\[
x^2 - x + y^2 - 2y = \frac{11}{4}
\][/tex]
3. Complete the square for the [tex]\( x \)[/tex] terms:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-1\)[/tex], half it, giving [tex]\(-\frac{1}{2}\)[/tex], and then square it, resulting in [tex]\(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex].
- Add and subtract [tex]\(\frac{1}{4}\)[/tex] within the [tex]\( x \)[/tex] terms.
[tex]\[
x^2 - x + \frac{1}{4} - \frac{1}{4}
\][/tex]
This converts [tex]\( x^2 - x + \frac{1}{4} \)[/tex] to [tex]\( (x - \frac{1}{2})^2 - \frac{1}{4} \)[/tex].
4. Complete the square for the [tex]\( y \)[/tex] terms:
- Take the coefficient of [tex]\( y \)[/tex], which is [tex]\(-2\)[/tex], half it, giving [tex]\(-1\)[/tex], and then square it, resulting in [tex]\((-1)^2 = 1\)[/tex].
- Add and subtract [tex]\( 1 \)[/tex] within the [tex]\( y \)[/tex] terms.
[tex]\[
y^2 - 2y + 1 - 1
\][/tex]
This converts [tex]\( y^2 - 2y + 1 \)[/tex] to [tex]\( (y - 1)^2 - 1 \)[/tex].
5. Rewrite the equation using the completed squares:
[tex]\[
(x - \frac{1}{2})^2 - \frac{1}{4} + (y - 1)^2 - 1 = \frac{11}{4}
\][/tex]
6. Combine constants on the right-hand side:
[tex]\[
(x - \frac{1}{2})^2 + (y - 1)^2 - \frac{5}{4} = \frac{11}{4}
\][/tex]
Move [tex]\(\frac{5}{4}\)[/tex] to the right-hand side:
[tex]\[
(x - \frac{1}{2})^2 + (y - 1)^2 = \frac{11}{4} + \frac{5}{4}
\][/tex]
Simplify the terms on the right-hand side:
[tex]\[
(x - \frac{1}{2})^2 + (y - 1)^2 = 4
\][/tex]
7. Identify the center and radius:
- The equation is now in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\( h = \frac{1}{2} \)[/tex], [tex]\( k = 1 \)[/tex], and [tex]\( r^2 = 4 \)[/tex].
- Thus, the center [tex]\((h, k) = (\frac{1}{2}, 1)\)[/tex].
- The radius [tex]\( r = \sqrt{4} = 2 \)[/tex].
### Conclusion
The coordinates for the center of the circle and the length of the radius are:
[tex]\[
\boxed{\left(\frac{1}{2}, 1\right), 2 \text{ units}}
\][/tex]
So the correct answer is Option D: [tex]\(\left(\frac{1}{2}, 1\right), 2 \text{ units}\)[/tex].
