Sure, let's solve this step-by-step.
1. General Form of a Circle’s Equation:
- The general form of a circle's equation 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. Given Center of the Circle:
- The center of the circle is given as [tex]\((-2, 1)\)[/tex].
3. Equation of the Circle:
- Plugging the center [tex]\((-2, 1)\)[/tex] into the equation, we get:
[tex]\[(x + 2)^2 + (y - 1)^2 = r^2\][/tex]
4. Expand the Equation:
- Expanding [tex]\((x + 2)^2 + (y - 1)^2\)[/tex]:
[tex]\[
(x + 2)^2 + (y - 1)^2 = x^2 + 4x + 4 + y^2 - 2y + 1
\][/tex]
[tex]\[
x^2 + y^2 + 4x - 2y + 5 = r^2
\][/tex]
5. General Form for Comparison:
- We need to compare the above equation with the given options to find which one matches.
6. Matching with Options:
- Option 1: [tex]\(x^2 + y^2 - 4x + 2y + 1 = 0\)[/tex]
This does not match the expanded form [tex]\(x^2 + y^2 + 4x - 2y + 5 = r^2\)[/tex].
- Option 2: [tex]\(x^2 + y^2 + 4x - 2y + 1 = 0\)[/tex]
Rewriting, we get:
[tex]\[
x^2 + y^2 + 4x - 2y + 1 = 0
\][/tex]
Comparing with [tex]\((x + 2)^2 + (y - 1)^2 = r^2 \Rightarrow x^2 + y^2 + 4x - 2y + 5 = r^2\)[/tex], we can conclude that
[tex]\(r^2 = 4\)[/tex],
since we need to change the constant term by subtracting 1 on both sides.
Thus, this option matches.
- Option 3: [tex]\(x^2 + y^2 + 4x - 2y + 9 = 0\)[/tex]
This does not match because the constant term is different when compared with the expanded form.
- Option 4: [tex]\(x^2 - y^2 + 2x + y + 1 = 0\)[/tex]
This is not even in the standard form of a circle's equation.
So, the correct form of the equation of the circle with center [tex]\((-2, 1)\)[/tex] is:
[tex]\[
x^2 + y^2 + 4x - 2y + 1 = 0
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
