Answer :
To find 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 can follow these steps:
1. Rewrite the equation in the standard form: We start by completing the square for both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
2. Complete the square for [tex]\(x\)[/tex]:
- Rewrite [tex]\(x^2 - x\)[/tex].
- To complete the square, take half the coefficient of [tex]\(x\)[/tex], which is [tex]\(-1/2\)[/tex], square it to get [tex]\((1/2)^2 = 1/4\)[/tex], and then add and subtract this square inside the equation.
[tex]\[ x^2 - x = (x - \frac{1}{2})^2 - \frac{1}{4} \][/tex]
3. Complete the square for [tex]\(y\)[/tex]:
- Rewrite [tex]\(y^2 - 2y\)[/tex].
- To complete the square, take half the coefficient of [tex]\(y\)[/tex], which is [tex]\(-2/2 = -1\)[/tex], square it to get [tex]\((-1)^2 = 1\)[/tex], and then add and subtract this square inside the equation.
[tex]\[ y^2 - 2y = (y - 1)^2 - 1 \][/tex]
4. Substitute these completed squares back into the circle's equation:
[tex]\[ (x - \frac{1}{2})^2 - \frac{1}{4} + (y - 1)^2 - 1 - \frac{11}{4} = 0 \][/tex]
We can simplify this equation by combining all the constants on the right side:
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 - \frac{15}{4} = 0 \][/tex]
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 = \frac{15}{4} \][/tex]
Now we have the equation in the standard form [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.
From the equation [tex]\((x - \frac{1}{2})^2 + (y - 1)^2 = \frac{15}{4}\)[/tex], we can see:
- The center [tex]\((h, k)\)[/tex] is [tex]\(\left(\frac{1}{2}, 1\right)\)[/tex].
- The radius squared [tex]\((r^2)\)[/tex] is [tex]\(\frac{15}{4}\)[/tex], so the radius [tex]\(r\)[/tex] is:
[tex]\[ r = \sqrt{\frac{15}{4}} \approx 1.936 \][/tex]
Thus, the coordinates for the center of the circle are [tex]\((\frac{1}{2}, 1)\)[/tex] and the radius is approximately [tex]\(1.936\)[/tex] units.
Therefore, the correct answer is neither of the provided options since the radius doesn't match exactly with the given choices. However, we can closely approximate the result to check for rounding. Given the closest fit in the context, we choose:
D. [tex]\(\left(\frac{1}{2}, 1\right), 4\)[/tex] units is not correct based on the calculated radius, but it can convey the idea if rounded or given without precise units. The correct pair would more accurately be aligned with correct numerical values.
Adjusting:
D. should have been [tex]\(\left(\frac{1}{2}, 1\right)\)[/tex] and radius ≈ [tex]\(1.936\)[/tex].
[tex]\[ x^2 + y^2 - x - 2y - \frac{11}{4} = 0 \][/tex]
we can follow these steps:
1. Rewrite the equation in the standard form: We start by completing the square for both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
2. Complete the square for [tex]\(x\)[/tex]:
- Rewrite [tex]\(x^2 - x\)[/tex].
- To complete the square, take half the coefficient of [tex]\(x\)[/tex], which is [tex]\(-1/2\)[/tex], square it to get [tex]\((1/2)^2 = 1/4\)[/tex], and then add and subtract this square inside the equation.
[tex]\[ x^2 - x = (x - \frac{1}{2})^2 - \frac{1}{4} \][/tex]
3. Complete the square for [tex]\(y\)[/tex]:
- Rewrite [tex]\(y^2 - 2y\)[/tex].
- To complete the square, take half the coefficient of [tex]\(y\)[/tex], which is [tex]\(-2/2 = -1\)[/tex], square it to get [tex]\((-1)^2 = 1\)[/tex], and then add and subtract this square inside the equation.
[tex]\[ y^2 - 2y = (y - 1)^2 - 1 \][/tex]
4. Substitute these completed squares back into the circle's equation:
[tex]\[ (x - \frac{1}{2})^2 - \frac{1}{4} + (y - 1)^2 - 1 - \frac{11}{4} = 0 \][/tex]
We can simplify this equation by combining all the constants on the right side:
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 - \frac{15}{4} = 0 \][/tex]
[tex]\[ (x - \frac{1}{2})^2 + (y - 1)^2 = \frac{15}{4} \][/tex]
Now we have the equation in the standard form [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.
From the equation [tex]\((x - \frac{1}{2})^2 + (y - 1)^2 = \frac{15}{4}\)[/tex], we can see:
- The center [tex]\((h, k)\)[/tex] is [tex]\(\left(\frac{1}{2}, 1\right)\)[/tex].
- The radius squared [tex]\((r^2)\)[/tex] is [tex]\(\frac{15}{4}\)[/tex], so the radius [tex]\(r\)[/tex] is:
[tex]\[ r = \sqrt{\frac{15}{4}} \approx 1.936 \][/tex]
Thus, the coordinates for the center of the circle are [tex]\((\frac{1}{2}, 1)\)[/tex] and the radius is approximately [tex]\(1.936\)[/tex] units.
Therefore, the correct answer is neither of the provided options since the radius doesn't match exactly with the given choices. However, we can closely approximate the result to check for rounding. Given the closest fit in the context, we choose:
D. [tex]\(\left(\frac{1}{2}, 1\right), 4\)[/tex] units is not correct based on the calculated radius, but it can convey the idea if rounded or given without precise units. The correct pair would more accurately be aligned with correct numerical values.
Adjusting:
D. should have been [tex]\(\left(\frac{1}{2}, 1\right)\)[/tex] and radius ≈ [tex]\(1.936\)[/tex].
