To determine the radius of the circle given by the equation [tex]\( x^2 + y^2 + 8x - 6y + 21 = 0 \)[/tex], let's rewrite the equation in the standard form [tex]\( (x-h)^2 + (y-k)^2 = r^2 \)[/tex]. This involves completing the square for the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms. Here are the detailed steps:
1. Complete the square for the [tex]\( x \)[/tex] terms:
The terms involving [tex]\( x \)[/tex] are [tex]\( x^2 + 8x \)[/tex].
- Take the coefficient of [tex]\( x \)[/tex], which is 8. Divide it by 2 to get 4, then square 4 to get 16.
- Add and subtract 16 to/from the [tex]\( x \)[/tex] terms:
[tex]\[
x^2 + 8x = (x + 4)^2 - 16
\][/tex]
2. Complete the square for the [tex]\( y \)[/tex] terms:
The terms involving [tex]\( y \)[/tex] are [tex]\( y^2 - 6y \)[/tex].
- Take the coefficient of [tex]\( y \)[/tex], which is -6. Divide it by 2 to get -3, then square -3 to get 9.
- Add and subtract 9 to/from the [tex]\( y \)[/tex] terms:
[tex]\[
y^2 - 6y = (y - 3)^2 - 9
\][/tex]
3. Rewrite the original equation with the completed squares:
Substitute the completed square forms back into the equation:
[tex]\[
(x + 4)^2 - 16 + (y - 3)^2 - 9 + 21 = 0
\][/tex]
4. Simplify the equation:
Combine the constants:
[tex]\[
(x + 4)^2 - 16 + (y - 3)^2 - 9 + 21 = (x + 4)^2 + (y - 3)^2 - 4 = 0
\][/tex]
5. Rearrange into the standard form of a circle:
[tex]\[
(x + 4)^2 + (y - 3)^2 = 4
\][/tex]
6. Identify the radius:
From the standard form [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex], we can see that [tex]\( r^2 = 4 \)[/tex]. Thus, [tex]\( r = \sqrt{4} = 2 \)[/tex].
Therefore, the radius of the circle is 2 units.
