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The general form of the equation of a circle is [tex]$x^2+y^2+42x+38y-47=0$[/tex]. The equation of this circle in standard form is [tex]\square[/tex].

The center of the circle is at the point [tex]\square[/tex], and its radius is [tex]\square[/tex] units.

The general form of the equation of a circle that has the same radius as the above circle is [tex]\square[/tex].



Answer :

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]