To find the center and the radius of the circle represented by the equation [tex]\(x^2 + 4x + y^2 - 2y - 4 = 0\)[/tex], we need to rewrite this equation in the standard form of a circle equation [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
Here's a detailed step-by-step approach:
1. Complete the Square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
- For [tex]\(x\)[/tex], we start with [tex]\(x^2 + 4x\)[/tex]:
[tex]\[
x^2 + 4x = (x + 2)^2 - 4
\][/tex]
- For [tex]\(y\)[/tex], we start with [tex]\(y^2 - 2y\)[/tex]:
[tex]\[
y^2 - 2y = (y - 1)^2 - 1
\][/tex]
2. Rewrite the original equation by including these completed squares:
[tex]\[
(x + 2)^2 - 4 + (y - 1)^2 - 1 - 4 = 0
\][/tex]
3. Combine and simplify the equation:
[tex]\[
(x + 2)^2 + (y - 1)^2 - 9 = 0
\][/tex]
4. Move the constant term to the other side to get the equation in standard form:
[tex]\[
(x + 2)^2 + (y - 1)^2 = 9
\][/tex]
Now we can identify the center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] of the circle:
- The center [tex]\((h, k)\)[/tex] is [tex]\((-2, 1)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Thus, the correct features of the circle are:
- The center is at [tex]\((-2, 1)\)[/tex].
- The radius is [tex]\(3\)[/tex].
The correct options are:
- radius of 3
- center at [tex]$(-2,1)$[/tex]
