At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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].
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].
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
