To determine the general form of the equation of a circle with center [tex]\((-2, 1)\)[/tex] that passes through the point [tex]\((-4, 1)\)[/tex], we can follow a series of steps.
### Step-by-Step Solution:
1. Circle’s Standard Form:
The standard form of the equation of a circle is:
[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.
2. Identify the Center:
Given the center [tex]\((-2, 1)\)[/tex], we have:
[tex]\[
h = -2 \quad \text{and} \quad k = 1
\][/tex]
3. Find the Radius:
The radius [tex]\(r\)[/tex] is the distance between the center [tex]\((-2, 1)\)[/tex] and the given point [tex]\((-4, 1)\)[/tex]. The distance formula is:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the given points:
[tex]\[
r = \sqrt{(-4 - (-2))^2 + (1 - 1)^2} = \sqrt{(-4 + 2)^2} = \sqrt{(-2)^2} = \sqrt{4} = 2
\][/tex]
4. Substitute Center and Radius into Standard Form:
Substituting [tex]\( h = -2 \)[/tex], [tex]\( k = 1 \)[/tex], and [tex]\( r = 2 \)[/tex] into the standard form:
[tex]\[
(x + 2)^2 + (y - 1)^2 = 4
\][/tex]
5. Expand the Equation:
[tex]\[
(x + 2)^2 = x^2 + 4x + 4
\][/tex]
[tex]\[
(y - 1)^2 = y^2 - 2y + 1
\][/tex]
Putting these together:
[tex]\[
x^2 + 4x + 4 + y^2 - 2y + 1 = 4
\][/tex]
6. Simplify:
Combine like terms:
[tex]\[
x^2 + y^2 + 4x - 2y + 5 = 4
\][/tex]
Subtract [tex]\(4\)[/tex] from both sides to get the general form:
[tex]\[
x^2 + y^2 + 4x - 2y + 1 = 0
\][/tex]
### Conclusion:
The correct answer is:
[tex]\[
A. \quad x^2 + y^2 + 4x - 2y + 1 = 0
\][/tex]
