To determine the radius of the circle given by the equation [tex]\( z^2 + y^2 + 6x - 8y - 10 = 0 \)[/tex], we need to rewrite it 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 and [tex]\(r\)[/tex] is the radius.
Let's complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
1. Rearrange the equation for completing the square:
[tex]\[
x^2 + y^2 + 6x - 8y - 10 = 0
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
- The relevant terms are [tex]\(x^2\)[/tex] and [tex]\(6x\)[/tex].
- Take half the coefficient of [tex]\(x\)[/tex] (which is 6), square it, and add/subtract that inside the equation:
[tex]\[
x^2 + 6x \quad \Rightarrow \quad (x + 3)^2 - 9
\][/tex]
- Here, [tex]\((x + 3)^2 - 9\)[/tex] is the completed square form.
3. Complete the square for the [tex]\(y\)[/tex]-terms:
- The relevant terms are [tex]\(y^2\)[/tex] and [tex]\(-8y\)[/tex].
- Take half the coefficient of [tex]\(y\)[/tex] (which is -8), square it, and add/subtract that inside the equation:
[tex]\[
y^2 - 8y \quad \Rightarrow \quad (y - 4)^2 - 16
\][/tex]
- Here, [tex]\((y - 4)^2 - 16\)[/tex] is the completed square form.
4. Substitute these back into the original equation:
[tex]\[
(x + 3)^2 - 9 + (y - 4)^2 - 16 - 10 = 0
\][/tex]
5. Simplify the equation:
[tex]\[
(x + 3)^2 + (y - 4)^2 - 35 = 0
\][/tex]
[tex]\[
(x + 3)^2 + (y - 4)^2 = 35
\][/tex]
From this equation, [tex]\((x + 3)^2 + (y - 4)^2 = 35\)[/tex], we can see that the circle has its center at [tex]\((-3, 4)\)[/tex] and its radius squared is equal to 35.
Therefore, the radius [tex]\(r\)[/tex] is:
[tex]\[
r = \sqrt{35}
\][/tex]
The correct answer is [tex]\(\sqrt{35}\)[/tex] units.
