To convert the given equation of the circle from general form to standard form, follow these steps:
1. Start with the general form of the circle's equation:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = 0 \][/tex]
2. Group the x-terms and y-terms together:
[tex]\[ (x^2 + 8x) + (y^2 + 22y) + 37 = 0 \][/tex]
3. Complete the square for the x-terms and y-terms:
For [tex]\(x^2 + 8x\)[/tex]:
- Take the coefficient of x, which is 8, and divide it by 2, giving 4.
- Square this value: [tex]\(4^2 = 16\)[/tex].
- Rewrite [tex]\(x^2 + 8x\)[/tex] as [tex]\((x + 4)^2 - 16\)[/tex].
For [tex]\(y^2 + 22y\)[/tex]:
- Take the coefficient of y, which is 22, and divide it by 2, giving 11.
- Square this value: [tex]\(11^2 = 121\)[/tex].
- Rewrite [tex]\(y^2 + 22y\)[/tex] as [tex]\((y + 11)^2 - 121\)[/tex].
Thus, the equation now becomes:
[tex]\[ (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 = 0 \][/tex]
4. Combine like terms:
[tex]\[ (x + 4)^2 + (y + 11)^2 - 100 = 0 \][/tex]
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
So, the standard form of the equation is:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
5. Identify the circle's center and radius squared:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-4, -11)\)[/tex].
- The radius squared [tex]\(r^2\)[/tex] is [tex]\(100\)[/tex].
Therefore, the complete answers are:
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]
