Answer :
To determine the length of the radius of the circle represented by the equation [tex]\((x+6)^2+(y+8)^2=9\)[/tex], we can use the standard form of the equation of a circle. The general form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] represents the radius.
1. Identify the center and radius:
- From the given equation [tex]\((x + 6)^2 + (y + 8)^2 = 9\)[/tex], we can observe that:
- [tex]\((x + 6)^2\)[/tex] can be rewritten as [tex]\((x - (-6))^2\)[/tex]. Thus, [tex]\(h = -6\)[/tex].
- [tex]\((y + 8)^2\)[/tex] can be rewritten as [tex]\((y - (-8))^2\)[/tex]. Thus, [tex]\(k = -8\)[/tex].
Therefore, the center of the circle is [tex]\((-6, -8)\)[/tex].
2. Determine the radius:
- The right side of the equation, [tex]\(9\)[/tex], represents [tex]\(r^2\)[/tex] (the square of the radius).
- To find the radius [tex]\(r\)[/tex], we need to take the square root of [tex]\(9\)[/tex].
3. Calculate the radius:
[tex]\[
r = \sqrt{9} = 3
\][/tex]
Thus, the length of the radius of circle [tex]\(F\)[/tex] is [tex]\(3\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] represents the radius.
1. Identify the center and radius:
- From the given equation [tex]\((x + 6)^2 + (y + 8)^2 = 9\)[/tex], we can observe that:
- [tex]\((x + 6)^2\)[/tex] can be rewritten as [tex]\((x - (-6))^2\)[/tex]. Thus, [tex]\(h = -6\)[/tex].
- [tex]\((y + 8)^2\)[/tex] can be rewritten as [tex]\((y - (-8))^2\)[/tex]. Thus, [tex]\(k = -8\)[/tex].
Therefore, the center of the circle is [tex]\((-6, -8)\)[/tex].
2. Determine the radius:
- The right side of the equation, [tex]\(9\)[/tex], represents [tex]\(r^2\)[/tex] (the square of the radius).
- To find the radius [tex]\(r\)[/tex], we need to take the square root of [tex]\(9\)[/tex].
3. Calculate the radius:
[tex]\[
r = \sqrt{9} = 3
\][/tex]
Thus, the length of the radius of circle [tex]\(F\)[/tex] is [tex]\(3\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
