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Sagot :
First, we need to convert the given equation of the circle from general form to standard form. The given equation is:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
We will complete the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
### Step-by-Step Conversion:
1. Rearrange the equation to group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[ (x^2 + 8x) + (y^2 + 22y) = -37 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 + 8x \][/tex]
To complete the square, we take half of the coefficient of [tex]\(x\)[/tex], which is 8, divide by 2 to get 4, and then square it, which gives us 16. Add and subtract 16 within the terms:
[tex]\[ x^2 + 8x + 16 - 16 \][/tex]
This simplifies to:
[tex]\[ (x + 4)^2 - 16 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 + 22y \][/tex]
Similarly, we take half of the coefficient of [tex]\(y\)[/tex], which is 22, divide by 2 to get 11, and then square it, which gives us 121. Add and subtract 121 within the terms:
[tex]\[ y^2 + 22y + 121 - 121 \][/tex]
This simplifies to:
[tex]\[ (y + 11)^2 - 121 \][/tex]
4. Combine the completed squares and set the equation equal to the constants:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 = -37 \][/tex]
5. Simplify the constants on the right-hand side:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 137 = -37 \][/tex]
Adding 137 to both sides:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Standard Form:
So, the equation of the circle in standard form is:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Center and Radius:
The standard form of a circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
From the equation [tex]\((x + 4)^2 + (y + 11)^2 = 100\)[/tex], we have:
- The center is at [tex]\((-4, -11)\)[/tex]
- The radius is [tex]\(\sqrt{100} = 10\)[/tex]
### Final Answers:
1. The equation of this circle in standard form is:
[tex]\((x + 4)^2 + (y + 11)^2 = 100\)[/tex]
2. The center of the circle is at the point:
[tex]\((-4, -11)\)[/tex]
Therefore:
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]\((x + \boxed{4})^2 + (y + \boxed{11})^2 = \boxed{100}\)[/tex].
The center of the circle is at the point [tex]\((\boxed{-4}, \boxed{-11})\)[/tex].
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
We will complete the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
### Step-by-Step Conversion:
1. Rearrange the equation to group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[ (x^2 + 8x) + (y^2 + 22y) = -37 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 + 8x \][/tex]
To complete the square, we take half of the coefficient of [tex]\(x\)[/tex], which is 8, divide by 2 to get 4, and then square it, which gives us 16. Add and subtract 16 within the terms:
[tex]\[ x^2 + 8x + 16 - 16 \][/tex]
This simplifies to:
[tex]\[ (x + 4)^2 - 16 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 + 22y \][/tex]
Similarly, we take half of the coefficient of [tex]\(y\)[/tex], which is 22, divide by 2 to get 11, and then square it, which gives us 121. Add and subtract 121 within the terms:
[tex]\[ y^2 + 22y + 121 - 121 \][/tex]
This simplifies to:
[tex]\[ (y + 11)^2 - 121 \][/tex]
4. Combine the completed squares and set the equation equal to the constants:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 = -37 \][/tex]
5. Simplify the constants on the right-hand side:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 137 = -37 \][/tex]
Adding 137 to both sides:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Standard Form:
So, the equation of the circle in standard form is:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Center and Radius:
The standard form of a circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
From the equation [tex]\((x + 4)^2 + (y + 11)^2 = 100\)[/tex], we have:
- The center is at [tex]\((-4, -11)\)[/tex]
- The radius is [tex]\(\sqrt{100} = 10\)[/tex]
### Final Answers:
1. The equation of this circle in standard form is:
[tex]\((x + 4)^2 + (y + 11)^2 = 100\)[/tex]
2. The center of the circle is at the point:
[tex]\((-4, -11)\)[/tex]
Therefore:
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]\((x + \boxed{4})^2 + (y + \boxed{11})^2 = \boxed{100}\)[/tex].
The center of the circle is at the point [tex]\((\boxed{-4}, \boxed{-11})\)[/tex].
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