At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Ask your questions and receive precise answers from experienced professionals across different disciplines. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the center of the circle given by the equation [tex]\( x^2 + y^2 + 4x - 8y + 11 = 0 \)[/tex], we need to rewrite the equation in the standard form of a circle's equation, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
1. Rewrite the given equation and group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[ x^2 + y^2 + 4x - 8y + 11 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
[tex]\[ x^2 + 4x \][/tex]
To complete the square, add and subtract [tex]\(4\)[/tex] inside the equation:
[tex]\[ x^2 + 4x + 4 - 4 = 0 \][/tex]
Factorize the trinomial:
[tex]\[ (x + 2)^2 - 4 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex]-terms:
[tex]\[ y^2 - 8y \][/tex]
To complete the square, add and subtract [tex]\(16\)[/tex] inside the equation:
[tex]\[ y^2 - 8y + 16 - 16 = 0 \][/tex]
Factorize the trinomial:
[tex]\[ (y - 4)^2 - 16 \][/tex]
4. Substitute the completed squares back into the original equation:
[tex]\[ (x + 2)^2 - 4 + (y - 4)^2 - 16 + 11 = 0 \][/tex]
5. Simplify the equation:
[tex]\[ (x + 2)^2 + (y - 4)^2 - 9 = 0 \][/tex]
[tex]\[ (x + 2)^2 + (y - 4)^2 = 9 \][/tex]
Now, the equation is in the standard form, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\( h = -2 \)[/tex] and [tex]\( k = 4 \)[/tex].
Therefore, the center of the circle is [tex]\((-2, 4)\)[/tex].
The correct answer is:
[tex]\((-2, 4)\)[/tex]
1. Rewrite the given equation and group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[ x^2 + y^2 + 4x - 8y + 11 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
[tex]\[ x^2 + 4x \][/tex]
To complete the square, add and subtract [tex]\(4\)[/tex] inside the equation:
[tex]\[ x^2 + 4x + 4 - 4 = 0 \][/tex]
Factorize the trinomial:
[tex]\[ (x + 2)^2 - 4 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex]-terms:
[tex]\[ y^2 - 8y \][/tex]
To complete the square, add and subtract [tex]\(16\)[/tex] inside the equation:
[tex]\[ y^2 - 8y + 16 - 16 = 0 \][/tex]
Factorize the trinomial:
[tex]\[ (y - 4)^2 - 16 \][/tex]
4. Substitute the completed squares back into the original equation:
[tex]\[ (x + 2)^2 - 4 + (y - 4)^2 - 16 + 11 = 0 \][/tex]
5. Simplify the equation:
[tex]\[ (x + 2)^2 + (y - 4)^2 - 9 = 0 \][/tex]
[tex]\[ (x + 2)^2 + (y - 4)^2 = 9 \][/tex]
Now, the equation is in the standard form, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\( h = -2 \)[/tex] and [tex]\( k = 4 \)[/tex].
Therefore, the center of the circle is [tex]\((-2, 4)\)[/tex].
The correct answer is:
[tex]\((-2, 4)\)[/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.