A circle is defined by the equation given below:

[tex]\[ x^2 + y^2 - x - 2y - \frac{11}{4} = 0 \][/tex]

What are the coordinates of the center of the circle and the length of the radius?

A. [tex]\(\left(-\frac{1}{2}, -1\right), 2\)[/tex] units
B. [tex]\(\left(\frac{1}{2}, 1\right), 4\)[/tex] units
C. [tex]\(\left(\frac{1}{2}, 1\right), 2\)[/tex] units
D. [tex]\(\left(-\frac{1}{2}, -1\right), 4\)[/tex] units



Answer :

To find the coordinates of the center and the radius of the circle defined by the equation:
[tex]\[ x^2 + y^2 - x - 2y - \frac{11}{4} = 0, \][/tex]
we need to rewrite it in the standard form of a circle's equation, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].

Let’s proceed step by step:

Step 1: Group the x and y terms.

The given equation is:
[tex]\[ x^2 + y^2 - x - 2y - \frac{11}{4} = 0. \][/tex]

Group the [tex]\(x\)[/tex] terms together and the [tex]\(y\)[/tex] terms together:
[tex]\[ (x^2 - x) + (y^2 - 2y) = \frac{11}{4}. \][/tex]

Step 2: Complete the square for the [tex]\(x\)[/tex] terms and [tex]\(y\)[/tex] terms.

For the [tex]\(x\)[/tex] terms ([tex]\(x^2 - x\)[/tex]):
- Take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-1/2\)[/tex], square it, and add and subtract it inside the group. This gives us [tex]\((x - \frac{1}{2})^2 - (\frac{1}{2})^2\)[/tex].

For the [tex]\(y\)[/tex] terms ([tex]\(y^2 - 2y\)[/tex]):
- Take half of the coefficient of [tex]\(y\)[/tex], which is [tex]\(-1\)[/tex], square it, and add and subtract it inside the group. This gives us [tex]\((y - 1)^2 - 1\)[/tex].

Substitute these completed square terms back into the equation:
[tex]\[ (x - \frac{1}{2})^2 - (\frac{1}{2})^2 + (y - 1)^2 - 1 = \frac{11}{4}. \][/tex]

Step 3: Simplify the equation.

Combine the constant terms on the left-hand side:
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 - \frac{1}{4} - 1 = \frac{11}{4}. \][/tex]

Move the constants to the right-hand side:
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 = \frac{11}{4} + \frac{1}{4} + 1. \][/tex]

Combine the constants on the right-hand side:
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 = \frac{16}{4} = 4. \][/tex]

Step 4: Identify the center and the radius.

Now, the equation is in the standard form:
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 = 4. \][/tex]

From this equation, we can see that:
- The center [tex]\((h, k)\)[/tex] is [tex]\(\left(\frac{1}{2}, 1\right)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{4} = 2\)[/tex].

Conclusion:

The coordinates for the center of the circle are [tex]\(\left(\frac{1}{2}, 1\right)\)[/tex], and the radius is [tex]\(2\)[/tex] units. Therefore, the correct answer is:

C. [tex]\(\left(\frac{1}{2}, 1\right), 2\)[/tex] units.