To find the radius of the circle given by the equation [tex]\(x^2 + y^2 + 6x - 8y - 10 = 0\)[/tex], we need to rewrite this equation in the standard form of a circle's equation: [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
1. Group the [tex]\(x\)[/tex] terms and [tex]\(y\)[/tex] terms:
[tex]\[
(x^2 + 6x) + (y^2 - 8y) = 10
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
- Take the coefficient of [tex]\(x\)[/tex], which is 6, halve it to get 3, and then square it to get [tex]\(3^2 = 9\)[/tex].
- Add and subtract this square inside the equation:
[tex]\[
x^2 + 6x + 9 - 9 = (x + 3)^2 - 9
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
- Take the coefficient of [tex]\(-8\)[/tex], halve it to get -4, and then square it to get [tex]\((-4)^2 = 16\)[/tex].
- Add and subtract this square inside the equation:
[tex]\[
y^2 - 8y + 16 - 16 = (y - 4)^2 - 16
\][/tex]
4. Rewrite the equation with the completed squares:
[tex]\[
(x^2 + 6x + 9) + (y^2 - 8y + 16) = 10 + 9 + 16
\][/tex]
Simplify to get:
[tex]\[
(x + 3)^2 + (y - 4)^2 = 35
\][/tex]
5. Identify the radius from the standard form:
The equation [tex]\((x + 3)^2 + (y - 4)^2 = 35\)[/tex] is now in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k) = (-3, 4)\)[/tex] and [tex]\(r^2 = 35\)[/tex].
Therefore, the radius [tex]\(r\)[/tex] is:
[tex]\[
r = \sqrt{35}
\][/tex]
Thus, the radius of the circle is [tex]\(\sqrt{35}\)[/tex] units.
