To determine the radius of the circle given by the equation [tex]\( x^2 + y^2 + 8x - 6y + 21 = 0 \)[/tex], we need to convert this equation into the standard form of a circle's equation, which is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. Let's do this step-by-step by completing the square.
1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms and move the constant to the other side of the equation:
[tex]\[
x^2 + 8x + y^2 - 6y = -21
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
- Take the coefficient of [tex]\(x\)[/tex], which is 8, halve it to get 4, and then square it to get 16.
- Add and subtract this squared term inside the equation.
[tex]\[
x^2 + 8x \implies (x + 4)^2 - 16
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex]-terms:
- Take the coefficient of [tex]\(y\)[/tex], which is -6, halve it to get -3, and then square it to get 9.
- Add and subtract this squared term inside the equation.
[tex]\[
y^2 - 6y \implies (y - 3)^2 - 9
\][/tex]
4. Substitute these squares back into the original equation:
[tex]\[
(x + 4)^2 - 16 + (y - 3)^2 - 9 = -21
\][/tex]
5. Combine the constants and simplify:
[tex]\[
(x + 4)^2 + (y - 3)^2 - 25 = -21
\][/tex]
[tex]\[
(x + 4)^2 + (y - 3)^2 = -21 + 25
\][/tex]
[tex]\[
(x + 4)^2 + (y - 3)^2 = 4
\][/tex]
6. Identify the radius [tex]\(r\)[/tex] from the standard form [tex]\( (x - h)^2 + (y - k)^2 = r^2\)[/tex]:
In this case:
[tex]\[
r^2 = 4 \implies r = \sqrt{4} \implies r = 2
\][/tex]
Therefore, the radius of the circle is [tex]\( \boxed{2} \)[/tex] units.
