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Sagot :
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.
[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.
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