To identify the center and radius of the circle given by the equation [tex]\( x^2 + y^2 + 14x + 2y + 14 = 0 \)[/tex], we need to rewrite it in standard form, which is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
Here is the step-by-step solution:
1. Rewrite the given equation in terms of completing the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[
x^2 + y^2 + 14x + 2y + 14 = 0
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms ([tex]\(x^2 + 14x\)[/tex]):
[tex]\[
x^2 + 14x
\][/tex]
To complete the square, add and subtract [tex]\(49\)[/tex] (since [tex]\((14/2)^2 = 49\)[/tex]):
[tex]\[
x^2 + 14x + 49 - 49 = (x + 7)^2 - 49
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms ([tex]\(y^2 + 2y\)[/tex]):
[tex]\[
y^2 + 2y
\][/tex]
To complete the square, add and subtract [tex]\(1\)[/tex] (since [tex]\((2/2)^2 = 1\)[/tex]):
[tex]\[
y^2 + 2y + 1 - 1 = (y + 1)^2 - 1
\][/tex]
4. Substitute these expressions back into the original equation:
[tex]\[
(x + 7)^2 - 49 + (y + 1)^2 - 1 + 14 = 0
\][/tex]
5. Simplify the equation:
[tex]\[
(x + 7)^2 + (y + 1)^2 - 50 + 14 = 0
\][/tex]
[tex]\[
(x + 7)^2 + (y + 1)^2 - 36 = 0
\][/tex]
6. Isolate the squared terms:
[tex]\[
(x + 7)^2 + (y + 1)^2 = 36
\][/tex]
Now, the equation is in standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. From this, we can determine the center and radius:
- The center [tex]\((h, k)\)[/tex] is found by examining the expressions within the squares:
[tex]\[
(x + 7)^2 \rightarrow h = -7
\][/tex]
[tex]\[
(y + 1)^2 \rightarrow k = -1
\][/tex]
- The radius [tex]\(r\)[/tex] is obtained from the right-hand side of the equation:
[tex]\[
(x + 7)^2 + (y + 1)^2 = 36 \rightarrow r^2 = 36 \rightarrow r = \sqrt{36} = 6
\][/tex]
Therefore, the center of the circle is [tex]\((-7, -1)\)[/tex] and the radius is [tex]\(6\)[/tex] units.
The correct answer is:
D. [tex]\((-7, -1), 6\)[/tex] units
