Sure, let's go through the process step-by-step to answer the question:
1. Convert the general form to standard form:
The given equation of the circle is [tex]\(x^2 + y^2 + 42x + 38y - 47 = 0\)[/tex].
To convert to the standard form (which is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]), we complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
- For the [tex]\(x\)[/tex] terms: [tex]\(x^2 + 42x\)[/tex]:
[tex]\[
x^2 + 42x = (x + 21)^2 - 441
\][/tex]
- For the [tex]\(y\)[/tex] terms: [tex]\(y^2 + 38y\)[/tex]:
[tex]\[
y^2 + 38y = (y + 19)^2 - 361
\][/tex]
Substitute back into the equation:
[tex]\[
(x + 21)^2 - 441 + (y + 19)^2 - 361 - 47 = 0
\][/tex]
Combine the constants:
[tex]\[
(x + 21)^2 + (y + 19)^2 = 849
\][/tex]
Therefore, the equation in standard form is:
[tex]\[
(x + 21)^2 + (y + 19)^2 = 849
\][/tex]
2. Identify the center of the circle:
From the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can see that the center [tex]\((h, k)\)[/tex] of the circle is [tex]\(( -21, -19)\)[/tex].
3. Determine the radius:
The right-hand side of the standard form equation is [tex]\(r^2\)[/tex].
[tex]\[
r^2 = 849 \implies r = \sqrt{849} \approx 29.13760456866693
\][/tex]
4. General form of another circle with the same radius:
The general form of a circle's equation is [tex]\(x^2 + y^2 + Dx + Ey + F = 0\)[/tex]. To have the same radius [tex]\(\sqrt{849}\)[/tex], we use the same constant on the right side:
One possible general form with the same center values (but different [tex]\((D, E)\)[/tex]) could be:
[tex]\[
x^2 + y^2 + 42x + 38y + C
\][/tex]
So, putting it all together:
1. The equation of the circle in standard form is [tex]\(\boxed{(x + 21)^2 + (y + 19)^2 = 849}\)[/tex].
2. The center of the circle is at the point [tex]\(\boxed{(-21, -19)}\)[/tex].
3. Its radius is [tex]\(\boxed{29.13760456866693}\)[/tex] units.
4. The general form of the equation of a circle that has the same radius as the above circle is [tex]\(\boxed{x^2 + y^2 + 42x + 38y + C}\)[/tex].
