Let's re-examine each point mentioned in the question to determine whether it lies on the circle represented by the equation [tex]\((x + 5)^2 + (y - 9)^2 = 8^2\)[/tex].
### Given Circle Equation
The equation of the circle is:
[tex]\[
(x + 5)^2 + (y - 9)^2 = 8^2
\][/tex]
### Checking Each Point
1. Point [tex]\((0, 8)\)[/tex]
[tex]\[
(x, y) = (0, 8)
\][/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 8 \)[/tex] into the circle equation:
[tex]\[
(0 + 5)^2 + (8 - 9)^2 = 8^2
\][/tex]
[tex]\[
5^2 + (-1)^2 = 8^2
\][/tex]
[tex]\[
25 + 1 = 64
\][/tex]
[tex]\[
26 \neq 64
\][/tex]
So, point [tex]\((0, 8)\)[/tex] does not lie on the circle.
2. Point [tex]\((13, -9)\)[/tex]
[tex]\[
(x, y) = (13, -9)
\][/tex]
Substitute [tex]\( x = 13 \)[/tex] and [tex]\( y = -9 \)[/tex] into the circle equation:
[tex]\[
(13 + 5)^2 + (-9 - 9)^2 = 8^2
\][/tex]
[tex]\[
18^2 + (-18)^2 = 8^2
\][/tex]
[tex]\[
324 + 324 = 64
\][/tex]
[tex]\[
648 \neq 64
\][/tex]
So, point [tex]\((13, -9)\)[/tex] does not lie on the circle.
3. Point [tex]\((-5, 1)\)[/tex]
[tex]\[
(x, y) = (-5, 1)
\][/tex]
Substitute [tex]\( x = -5 \)[/tex] and [tex]\( y = 1 \)[/tex] into the circle equation:
[tex]\[
(-5 + 5)^2 + (1 - 9)^2 = 8^2
\][/tex]
[tex]\[
0^2 + (-8)^2 = 8^2
\][/tex]
[tex]\[
0 + 64 = 64
\][/tex]
[tex]\[
64 = 64
\][/tex]
So, point [tex]\((-5, 1)\)[/tex] lies on the circle.
4. Point [tex]\((3, 17)\)[/tex]
[tex]\[
(x, y) = (3, 17)
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 17 \)[/tex] into the circle equation:
[tex]\[
(3 + 5)^2 + (17 - 9)^2 = 8^2
\][/tex]
[tex]\[
8^2 + 8^2 = 8^2
\][/tex]
[tex]\[
64 + 64 = 64
\][/tex]
[tex]\[
128 \neq 64
\][/tex]
So, point [tex]\((3, 17)\)[/tex] does not lie on the circle.
### Conclusion
Among the provided points, the only point that lies on the circle is [tex]\((-5, 1)\)[/tex].
Therefore, the correct answer is:
C. [tex]\((-5, 1)\)[/tex]
