Certainly! Let's work through the steps to convert the given general form of the equation of a circle to its standard form and determine its center.
Given the general form of the equation of the circle:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
### Step 1: Completing the Square
First, let's complete the square for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
#### For [tex]\( x \)[/tex]:
[tex]\[ x^2 + 8x \][/tex]
To complete the square, we take the coefficient of [tex]\( x \)[/tex], which is [tex]\( 8 \)[/tex], divide by [tex]\( 2 \)[/tex], and square it:
[tex]\[ \left( \frac{8}{2} \right)^2 = 16 \][/tex]
Thus, we can rewrite [tex]\( x^2 + 8x \)[/tex] as:
[tex]\[ (x + 4)^2 - 16 \][/tex]
#### For [tex]\( y \)[/tex]:
[tex]\[ y^2 + 22y \][/tex]
Similarly, take the coefficient of [tex]\( y \)[/tex], which is [tex]\( 22 \)[/tex], divide by [tex]\( 2 \)[/tex], and square it:
[tex]\[ \left( \frac{22}{2} \right)^2 = 121 \][/tex]
Thus, we can rewrite [tex]\( y^2 + 22y \)[/tex] as:
[tex]\[ (y + 11)^2 - 121 \][/tex]
### Step 2: Rewrite the Equation
Rewrite the original equation using these completed squares:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 = 0 \][/tex]
Combine like terms:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 100 = 0 \][/tex]
To isolate the squared terms on one side:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Step 3: Identify the Center and Radius
The standard form of the equation of a circle is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
From our equation:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
We can see that [tex]\( h = -4 \)[/tex], [tex]\( k = -11 \)[/tex], and [tex]\( r^2 = 100 \)[/tex]. Thus, the center of the circle is [tex]\( (-4, -11) \)[/tex].
### Filling in the Blanks
Finally, let's type the correct answers into 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].
Therefore, we have:
- First box: [tex]\( 4 \)[/tex]
- Second box: [tex]\( 11 \)[/tex]
- Third box: [tex]\( 100 \)[/tex]
- Fourth box: [tex]\( -4 \)[/tex]
- Fifth box: [tex]\( -11 \)[/tex]
