To determine the radius of a circle given 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, which is [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex]. Here is the step-by-step process to convert the given equation to the standard form by completing the square:
1. Start with the given equation:
[tex]\[
x^2 + y^2 + 8x - 6y + 21 = 0
\][/tex]
2. Group the [tex]\(x\)[/tex] terms and the [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:
[tex]\[
x^2 + 8x \quad \text{can be written as} \quad (x + 4)^2 - 16
\][/tex]
Here, we add and subtract 16 to complete the square.
4. Complete the square for the [tex]\(y\)[/tex] terms:
[tex]\[
y^2 - 6y \quad \text{can be written as} \quad (y - 3)^2 - 9
\][/tex]
Here, we add and subtract 9 to complete the square.
5. Substitute the completed squares back into the equation:
[tex]\[
(x + 4)^2 - 16 + (y - 3)^2 - 9 + 21 = 0
\][/tex]
6. Simplify the equation:
[tex]\[
(x + 4)^2 + (y - 3)^2 - 4 = 0
\][/tex]
By combining [tex]\(-16\)[/tex], [tex]\(-9\)[/tex], and [tex]\(21\)[/tex], we get:
[tex]\[
-16 - 9 + 21 = -4
\][/tex]
7. Isolate the completed square terms:
[tex]\[
(x + 4)^2 + (y - 3)^2 = 4
\][/tex]
Now, the equation is in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\(h = -4\)[/tex], [tex]\(k = 3\)[/tex], and [tex]\(r^2 = 4\)[/tex]. Thus, the radius [tex]\(r\)[/tex] is the square root of 4, which is 2 units.
Therefore, the radius of the circle is [tex]\(2\)[/tex] units.
