To find the center and radius of the circle given by the equation [tex]\(x^2 + y^2 + Ax + By + C = 0\)[/tex], we can use the following steps:
1. Identify the coefficients: The coefficients from the circle equation are [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
2. Calculate the center coordinates:
The center of the circle can be determined using the formula for the center coordinates:
[tex]\[
\left(-\frac{A}{2}, -\frac{B}{2}\right)
\][/tex]
Substitute [tex]\(A = 1\)[/tex] and [tex]\(B = 1\)[/tex] into the formula:
[tex]\[
\left(-\frac{1}{2}, -\frac{1}{2}\right)
\][/tex]
Thus, the center [tex]\((h, k)\)[/tex] is [tex]\((-0.5, -0.5)\)[/tex].
3. Calculate the radius:
The radius [tex]\(r\)[/tex] of the circle can be computed using the formula:
[tex]\[
r = \sqrt{\left(-\frac{A}{2}\right)^2 + \left(-\frac{B}{2}\right)^2 - C}
\][/tex]
Substitute [tex]\(A = 1\)[/tex], [tex]\(B = 1\)[/tex], and [tex]\(C = -3\)[/tex] into the formula:
[tex]\[
r = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 - (-3)}
\][/tex]
Calculate the terms inside the square root:
[tex]\[
\left(-\frac{1}{2}\right)^2 = 0.25
\][/tex]
[tex]\[
\left(-\frac{1}{2}\right)^2 = 0.25
\][/tex]
[tex]\[
-(-3) = 3
\][/tex]
Combine these values:
[tex]\[
r = \sqrt{0.25 + 0.25 + 3} = \sqrt{3.5}
\][/tex]
4. Determine the simplified radius value:
Evaluating [tex]\(\sqrt{3.5}\)[/tex]:
[tex]\[
r \approx 1.8708286933869707
\][/tex]
Therefore, the center and radius of the circle are:
[tex]\[
\text{Center}: (-0.5, -0.5)
\][/tex]
[tex]\[
\text{Radius}: 1.8708286933869707
\][/tex]
These steps ensure we correctly find the center and radius for the given circle equation [tex]\(x^2 + y^2 + x + y - 3 = 0\)[/tex].
