To convert the given equation of the circle from its general form to standard form, we need to complete the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
Given equation:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
### Step 1: Complete the square for the [tex]\( x \)[/tex] terms
1. Identify the coefficient of [tex]\(x\)[/tex], which is 8.
2. Take half of this coefficient: [tex]\(\frac{8}{2} = 4\)[/tex].
3. Square this value: [tex]\(4^2 = 16\)[/tex].
4. Add and subtract this square inside the equation:
[tex]\[ x^2 + 8x = (x + 4)^2 - 16 \][/tex]
### Step 2: Complete the square for the [tex]\( y \)[/tex] terms
1. Identify the coefficient of [tex]\(y\)[/tex], which is 22.
2. Take half of this coefficient: [tex]\(\frac{22}{2} = 11\)[/tex].
3. Square this value: [tex]\(11^2 = 121\)[/tex].
4. Add and subtract this square inside the equation:
[tex]\[ y^2 + 22y = (y + 11)^2 - 121 \][/tex]
### Step 3: Rewrite the original equation using completed squares
1. Substitute the completed squares into the original equation:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 = 0 \][/tex]
2. Combine the constants:
[tex]\[ -16 - 121 + 37 = -100 \][/tex]
3. Rewrite the equation:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 100 = 0 \][/tex]
4. Move the constant term to the other side of the equation:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Step 4: Extract information
1. The equation is now in standard form:
[tex]\[ (x + \boxed{4})^2 + (y + \boxed{11})^2 = \boxed{100} \][/tex]
2. The center of the circle, [tex]\((h, k)\)[/tex], is [tex]\((-4, -11)\)[/tex].
3. The radius, [tex]\( r \)[/tex], is [tex]\(\sqrt{100} = 10\)[/tex] (although we're not asked for the radius in this part).
So, filling out the boxes:
- The equation of this circle in standard form is [tex]\((x + \boxed{4})^2 + (y + \boxed{11})^2 = \boxed{100}\)[/tex].
- The center of the circle is at the point [tex]\( (\boxed{-4}, \boxed{-11}) \)[/tex].
