Let's start with the given equation of the circle in general form:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
To convert this to the standard form, we need to complete the square for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Complete the square for [tex]\( x \)[/tex]:
[tex]\[ x^2 + 8x \][/tex]
To complete the square, add and subtract [tex]\( 16 \)[/tex] (since [tex]\((\frac{8}{2})^2 = 16\)[/tex]):
[tex]\[ x^2 + 8x = (x + 4)^2 - 16 \][/tex]
2. Complete the square for [tex]\( y \)[/tex]:
[tex]\[ y^2 + 22y \][/tex]
To complete the square, add and subtract [tex]\( 121 \)[/tex] (since [tex]\((\frac{22}{2})^2 = 121\)[/tex]):
[tex]\[ y^2 + 22y = (y + 11)^2 - 121 \][/tex]
3. Rewrite the original equation using the completed squares:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
Substitute the completed squares:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 = 0 \][/tex]
4. Combine the constants:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 100 = 0 \][/tex]
5. Isolate the squared terms:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
Therefore, the equation in standard form is:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
The center of the circle is at the point [tex]\( (-4, -11) \)[/tex].
The radius of the circle is:
[tex]\[ \sqrt{100} = 10 \][/tex]
So, filling in the boxes, we get:
The equation of this circle in standard form is:
[tex]\[ (x + \square)^2 + (y + \square)^2 = \square \][/tex]
with the boxes filled in as follows:
1. [tex]\(\square\)[/tex]: -4
2. [tex]\(\square\)[/tex]: -11
3. [tex]\(\square\)[/tex]: 100
The center of the circle is at the point:
[tex]\[ (\square, \square) \][/tex]
with the boxes filled in as follows:
1. [tex]\(\square\)[/tex]: -4
2. [tex]\(\square\)[/tex]: -11
