To convert the given general form equation of the circle to its standard form, we need to complete the square for the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
Given equation:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
### Completing the Square for the [tex]\( x \)[/tex] term:
The [tex]\( x \)[/tex] part of the equation is [tex]\( x^2 + 8x \)[/tex].
1. Take the coefficient of [tex]\( x \)[/tex] (which is 8), divide by 2, and square it:
[tex]\[
\left( \frac{8}{2} \right)^2 = 16
\][/tex]
2. Rewrite [tex]\( x^2 + 8x \)[/tex] as a perfect square trinomial:
[tex]\[
x^2 + 8x + 16 - 16 = (x + 4)^2 - 16
\][/tex]
### Completing the Square for the [tex]\( y \)[/tex] term:
The [tex]\( y \)[/tex] part of the equation is [tex]\( y^2 + 22y \)[/tex].
1. Take the coefficient of [tex]\( y \)[/tex] (which is 22), divide by 2, and square it:
[tex]\[
\left( \frac{22}{2} \right)^2 = 121
\][/tex]
2. Rewrite [tex]\( y^2 + 22y \)[/tex] as a perfect square trinomial:
[tex]\[
y^2 + 22y + 121 - 121 = (y + 11)^2 - 121
\][/tex]
### Rewrite the original equation by substituting back:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 = 0 \][/tex]
### Simplifying the constants:
Combine [tex]\( -16 \)[/tex], [tex]\( -121 \)[/tex], and [tex]\( 37 \)[/tex]:
[tex]\[ -16 - 121 + 37 = -100 \][/tex]
So, the equation becomes:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 100 = 0 \][/tex]
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
### Identify the center and the radius of the circle:
The standard form of the equation of a circle 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 can see:
- The center [tex]\((h, k)\)[/tex] is [tex]\((-4, -11)\)[/tex]
- The radius squared [tex]\( r^2 \)[/tex] is 100, so the radius [tex]\( r \)[/tex] is [tex]\( \sqrt{100} = 10 \)[/tex]
### Final Results:
- The standard form of the circle equation: [tex]\((x + 4)^2 + (y + 11)^2 = 100\)[/tex]
- The center of the circle: [tex]\((-4, -11)\)[/tex]
- The radius of the circle: [tex]\(\sqrt{100} = 10\)[/tex]
Therefore:
- 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].
