To convert the given equation of the circle from its general form to standard form, we follow these steps:
1. Start with the given equation:
[tex]\[
x^2 + y^2 + 8x + 22y + 37 = 0
\][/tex]
2. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms and move the constant to the other side:
[tex]\[
x^2 + 8x + y^2 + 22y = -37
\][/tex]
3. Complete the square for the [tex]\(x\)[/tex] terms and [tex]\(y\)[/tex] terms:
- For the [tex]\(x\)[/tex] terms:
[tex]\[
x^2 + 8x \implies (x+4)^2 - 16
\][/tex]
- For the [tex]\(y\)[/tex] terms:
[tex]\[
y^2 + 22y \implies (y+11)^2 - 121
\][/tex]
4. Rewrite the equation incorporating these completed squares:
[tex]\[
(x+4)^2 - 16 + (y+11)^2 - 121 = -37
\][/tex]
5. Combine like terms on the right-hand side:
[tex]\[
(x+4)^2 + (y+11)^2 - 137 = -37
\][/tex]
[tex]\[
(x+4)^2 + (y+11)^2 = 100
\][/tex]
6. Identify the standard form of the equation:
[tex]\[
(x + 4)^2 + (y + 11)^2 = 100
\][/tex]
In this form, [tex]\((x + h)^2 + (y + k)^2 = r^2\)[/tex], we can identify:
[tex]\[
h = -4, \quad k = -11, \quad \text{and} \quad r^2 = 100 \quad (\text{thus,} \quad r = 10)
\][/tex]
7. Determine the center of the circle:
[tex]\[
\text{Center} = (-4, -11)
\][/tex]
So, the standard form of the equation of the circle is:
[tex]\[
(x + 4)^2 + (y + 11)^2 = 100
\][/tex]
And the center of the circle is:
[tex]\[
(-4, -11)
\][/tex]
Therefore, the completed solution for the answer boxes is:
The general form of the equation of a circle is [tex]\(x^2 + y^2 + 8x + 22y + 37 = 0\)[/tex]. The equation of this circle 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].
