Type the correct answer in each box.

The general form of the equation of a circle is [tex][tex]$x^2+y^2+8x+22y+37=0$[/tex][/tex].

The equation of this circle in standard form is [tex][tex]$(x+\square)^2+(y+\square)^2=\square$[/tex][/tex].

The center of the circle is at the point [tex][tex]$(\square, \square)$[/tex][/tex].



Answer :

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].