To determine the coordinates of the center and the radius of the circle described 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's equation, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] represents the radius.
### Step 1: Rearrange and Complete the Square
1. Start with the given equation:
[tex]\[
x^2 + y^2 - x - 2y - \frac{11}{4} = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 + y^2 - x - 2y = \frac{11}{4}
\][/tex]
3. Complete the square for the [tex]\(x\)[/tex] terms:
[tex]\[
x^2 - x \quad \Rightarrow \quad x^2 - x + \left(\frac{1}{2}\right)^2 = \left(x - \frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2
\][/tex]
Adding and subtracting [tex]\(\left(\frac{1}{2}\right)^2\)[/tex] gives us:
[tex]\[
x^2 - x + \frac{1}{4} = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4}
\][/tex]
4. Complete the square for the [tex]\(y\)[/tex] terms:
[tex]\[
y^2 - 2y \quad \Rightarrow \quad y^2 - 2y + 1 = (y - 1)^2 - 1
\][/tex]
Adding and subtracting 1 gives us:
[tex]\[
y^2 - 2y + 1 = (y - 1)^2 - 1
\][/tex]
5. Substitute back into the equation:
[tex]\[
\left(x - \frac{1}{2}\right)^2 - \frac{1}{4} + (y - 1)^2 - 1 = \frac{11}{4}
\][/tex]
6. Combine the constant terms and simplify:
[tex]\[
\left(x - \frac{1}{2}\right)^2 + (y - 1)^2 - \frac{5}{4} = \frac{11}{4}
\][/tex]
Adding [tex]\(\frac{5}{4}\)[/tex] to both sides:
[tex]\[
\left(x - \frac{1}{2}\right)^2 + (y - 1)^2 = \frac{16}{4} = 4
\][/tex]
### Step 2: Identify the Center and Radius
1. The equation is now in standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where:
[tex]\[
(x - \frac{1}{2})^2 + (y - 1)^2 = 4
\][/tex]
2. From this form, the center [tex]\((h, k)\)[/tex] is:
[tex]\[
\left(\frac{1}{2}, 1\right)
\][/tex]
3. The radius [tex]\(r\)[/tex] is the square root of 4:
[tex]\[
r = \sqrt{4} = 2
\][/tex]
### Step 3: Select the Correct Option
Comparing this result to the provided options, the coordinates for the center and the radius are:
[tex]\[
\text{C.} \left(\frac{1}{2}, 1\right), 2 \text{ units}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
