To find the radius of the circle given by the equation [tex]\(x^2 + y^2 + 8x - 6y + 21 = 0\)[/tex], we need to rewrite the equation in the standard form of a circle's equation:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
We'll complete the square for both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
1. Start with the given equation:
[tex]\[
x^2 + y^2 + 8x - 6y + 21 = 0
\][/tex]
2. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms together:
[tex]\[
(x^2 + 8x) + (y^2 - 6y) + 21 = 0
\][/tex]
3. Complete the square for the [tex]\(x\)[/tex] terms. To complete the square for [tex]\(x^2 + 8x\)[/tex], add and subtract [tex]\((\frac{8}{2})^2 = 16\)[/tex]:
[tex]\[
x^2 + 8x + 16 - 16
\][/tex]
This simplifies to:
[tex]\[
(x + 4)^2 - 16
\][/tex]
4. Complete the square for the [tex]\(y\)[/tex] terms. To complete the square for [tex]\(y^2 - 6y\)[/tex], add and subtract [tex]\((\frac{-6}{2})^2 = 9\)[/tex]:
[tex]\[
y^2 - 6y + 9 - 9
\][/tex]
This simplifies to:
[tex]\[
(y - 3)^2 - 9
\][/tex]
5. Substitute these completed squares back into the equation:
[tex]\[
(x + 4)^2 - 16 + (y - 3)^2 - 9 + 21 = 0
\][/tex]
6. Simplify the equation by combining constants:
[tex]\[
(x + 4)^2 + (y - 3)^2 - 25 = 0
\][/tex]
Adding 25 to both sides of the equation:
[tex]\[
(x + 4)^2 + (y - 3)^2 = 25
\][/tex]
7. We can now see that the equation of the circle is in the standard form, where [tex]\( (h, k) = (-4, 3) \)[/tex] and [tex]\( r^2 = 25 \)[/tex].
8. To find the radius [tex]\( r \)[/tex], we take the square root of 25:
[tex]\[
r = \sqrt{25} = 5
\][/tex]
Therefore, the radius of the circle is [tex]\(\boxed{5}\)[/tex].
