Sure! Let's begin by rewriting the given equation [tex]\(x^2 + 2x + y^2 - 8y + 1 = 0\)[/tex] into its graphing form, which is typically the standard form of a circle or an ellipse.
First, we need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
### Step 1: Complete the Square
For the [tex]\(x\)[/tex] terms:
The given terms involving [tex]\(x\)[/tex] are [tex]\(x^2 + 2x\)[/tex]. To complete the square:
1. Take the coefficient of [tex]\(x\)[/tex] which is [tex]\(2\)[/tex], divide it by 2: [tex]\(\frac{2}{2} = 1\)[/tex].
2. Square the result: [tex]\(1^2 = 1\)[/tex].
So, we add and subtract [tex]\(1\)[/tex] to complete the square:
[tex]\[x^2 + 2x = (x + 1)^2 - 1.\][/tex]
For the [tex]\(y\)[/tex] terms:
The given terms involving [tex]\(y\)[/tex] are [tex]\(y^2 - 8y\)[/tex]. To complete the square:
1. Take the coefficient of [tex]\(y\)[/tex] which is [tex]\(-8\)[/tex], divide it by 2: [tex]\(\frac{-8}{2} = -4\)[/tex].
2. Square the result: [tex]\((-4)^2 = 16\)[/tex].
So, we add and subtract [tex]\(16\)[/tex] to complete the square:
[tex]\[y^2 - 8y = (y - 4)^2 - 16.\][/tex]
### Step 2: Substitute Back into the Original Equation
We substitute these completed squares back into the original equation:
[tex]\[x^2 + 2x + y^2 - 8y + 1 = 0\][/tex]
Rewriting gives us:
[tex]\[(x + 1)^2 - 1 + (y - 4)^2 - 16 + 1 = 0.\][/tex]
### Step 3: Simplify the Equation
Now we simplify the equation:
[tex]\[(x + 1)^2 - 1 + (y - 4)^2 - 16 + 1 = 0\][/tex]
[tex]\[(x + 1)^2 - 16 + (y - 4)^2 = 0\][/tex]
[tex]\[(x + 1)^2 + (y - 4)^2 - 16 = 0\][/tex]
[tex]\[(x + 1)^2 + (y - 4)^2 = 16\][/tex]
This final equation [tex]\((x + 1)^2 + (y - 4)^2 = 16\)[/tex] represents a circle with center [tex]\((-1, 4)\)[/tex] and radius [tex]\(\sqrt{16} = 4\)[/tex].
### Step 4: Compare with Given Options
Comparing this with the given options:
a. [tex]\((x - 3)^2 + (y - 2)^2 = 1\)[/tex]
b. [tex]\((x - 2)^2 + (y + 5)^2 = 11\)[/tex]
c. [tex]\((x + 1)^2 + (y - 4)^2 = 16\)[/tex]
d. [tex]\((x + 2)^2 + (y - 1)^2 = 9\)[/tex]
The correct match is:
c. [tex]\((x + 1)^2 + (y - 4)^2 = 16\)[/tex].
Thus, the correct answer is option c.
