Answered

Select all the correct answers.

Consider the circle represented by this equation:
[tex]\[ x^2 + 4x + y^2 - 2y - 4 = 0 \][/tex]

Which features are characteristics of the circle?

A. center at [tex]\((2, -1)\)[/tex]
B. center at [tex]\((-4, 2)\)[/tex]
C. radius of 3
D. radius of 1
E. radius of 9
F. center at [tex]\((-2, 1)\)[/tex]



Answer :

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]