To determine which point lies on the circle, we first need to rewrite the equation of the circle in a more familiar form. The equation [tex]\((x+5)^2 + (y-9)^2 = 8^2\)[/tex] represents a circle centered at [tex]\((-5, 9)\)[/tex] with a radius of [tex]\(8\)[/tex].
To find out if a point [tex]\((x, y)\)[/tex] lies on this circle, we substitute the coordinates of the point into the equation of the circle and check if the equation holds true.
Let's examine each point:
A. [tex]\((0, 8)\)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 8\)[/tex]:
[tex]\[
(0+5)^2 + (8-9)^2 = 5^2 + (-1)^2 = 25 + 1 = 26 \neq 64
\][/tex]
Point [tex]\((0, 8)\)[/tex] does not lie on the circle.
B. [tex]\((13, -9)\)[/tex]
Substitute [tex]\(x = 13\)[/tex] and [tex]\(y = -9\)[/tex]:
[tex]\[
(13+5)^2 + (-9-9)^2 = 18^2 + (-18)^2 = 324 + 324 = 648 \neq 64
\][/tex]
Point [tex]\((13, -9)\)[/tex] does not lie on the circle.
C. [tex]\((-5, 1)\)[/tex]
Substitute [tex]\(x = -5\)[/tex] and [tex]\(y = 1\)[/tex]:
[tex]\[
(-5+5)^2 + (1-9)^2 = 0^2 + (-8)^2 = 0 + 64 = 64
\][/tex]
Point [tex]\((-5, 1)\)[/tex] does lie on the circle.
D. [tex]\((3, 17)\)[/tex]
Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 17\)[/tex]:
[tex]\[
(3+5)^2 + (17-9)^2 = 8^2 + 8^2 = 64 + 64 = 128 \neq 64
\][/tex]
Point [tex]\((3, 17)\)[/tex] does not lie on the circle.
Hence, the correct answer is:
C. [tex]\((-5, 1)\)[/tex]
