To determine the center and the radius of the circle given by the equation:
[tex]\[ x^2 + 4x + y^2 - 6y = -4 \][/tex]
we will rewrite this 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 and [tex]\(r\)[/tex] is the radius.
Step 1: Completing the square for the [tex]\(x\)[/tex] terms
[tex]\[ x^2 + 4x \][/tex]
To complete the square, add and subtract [tex]\((\frac{4}{2})^2 = 4\)[/tex]:
[tex]\[ x^2 + 4x + 4 - 4 \][/tex]
[tex]\[ (x + 2)^2 - 4 \][/tex]
Step 2: Completing the square for the [tex]\(y\)[/tex] terms
[tex]\[ y^2 - 6y \][/tex]
To complete the square, add and subtract [tex]\((\frac{-6}{2})^2 = 9\)[/tex]:
[tex]\[ y^2 - 6y + 9 - 9 \][/tex]
[tex]\[ (y - 3)^2 - 9 \][/tex]
Step 3: Substitute the completed squares back into the original equation
[tex]\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = -4 \][/tex]
Step 4: Combine and simplify
[tex]\[ (x + 2)^2 + (y - 3)^2 - 13 = -4 \][/tex]
[tex]\[ (x + 2)^2 + (y - 3)^2 = 9 \][/tex]
Now, the equation is in the standard form [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.
The center [tex]\((h, k)\)[/tex] is:
[tex]\[ (h, k) = (-2, 3) \][/tex]
The radius [tex]\(r\)[/tex] is:
[tex]\[ r = \sqrt{9} = 3 \][/tex]
Therefore, the correct answers are:
- The center is located at [tex]\((-2, 3)\)[/tex] and the radius is 3.
None of the given options indicate a radius of 9, so we can eliminate the options that mention a radius of 9.
Thus, the correct choice is:
The center is located at [tex]\((-2, 3)\)[/tex] and the radius is 3.
