Let's convert the given equation of the circle from its general form to its standard form step-by-step.
### Given Equation:
[tex]\[ x^2 + y^2 + 42x + 38y - 47 = 0 \][/tex]
### 1. Completing the Square:
For the x terms:
[tex]\[ x^2 + 42x \][/tex]
To complete the square:
[tex]\[ x^2 + 42x = (x + 21)^2 - 21^2 \][/tex]
For the y terms:
[tex]\[ y^2 + 38y \][/tex]
To complete the square:
[tex]\[ y^2 + 38y = (y + 19)^2 - 19^2 \][/tex]
### 2. Rewriting the Equation:
Substituting the completed squares into the original equation:
[tex]\[ (x + 21)^2 - 21^2 + (y + 19)^2 - 19^2 - 47 = 0 \][/tex]
### 3. Combining Constants:
[tex]\[ (x + 21)^2 + (y + 19)^2 - 21^2 - 19^2 - 47 = 0 \][/tex]
Calculate the constants:
[tex]\[ 21^2 = 441 \][/tex]
[tex]\[ 19^2 = 361 \][/tex]
[tex]\[ - 21^2 - 19^2 - 47 = - 441 - 361 - 47 = -849 \][/tex]
So, the equation becomes:
[tex]\[ (x + 21)^2 + (y + 19)^2 - 849 = 0 \][/tex]
### 4. Standard Form:
[tex]\[ (x + 21)^2 + (y + 19)^2 = 849 \][/tex]
### 5. Center and Radius:
The standard form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
From our equation:
[tex]\[ (x + 21)^2 + (y + 19)^2 = 849 \][/tex]
- Center [tex]\((h, k) = (-21, -19)\)[/tex]
- Radius [tex]\( r = \sqrt{849} \)[/tex]
### 6. New General Form with Center at Origin:
A new circle centered at the origin [tex]\((0,0)\)[/tex] and having the same radius would have the general form:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Since [tex]\( r^2 = 849 \)[/tex]:
[tex]\[ x^2 + y^2 = 849 \][/tex]
Thus, the general form is:
[tex]\[ x^2 + y^2 - 849 = 0 \][/tex]
### Final Answers:
- The standard form of the equation is: [tex]\(\boxed{(x + 21)^2 + (y + 19)^2 = 849}\)[/tex]
- The center of the circle is: [tex]\(\boxed{(-21, -19)}\)[/tex]
- The radius of the circle is: [tex]\(\boxed{\sqrt{849}}\)[/tex] units
- The general form of the new circle with the same radius centered at the origin is: [tex]\(\boxed{x^2 + y^2 - 849 = 0}\)[/tex]
