To determine which point lies on the circle represented by the equation [tex]\((x + 5)^2 + (y - 9)^2 = 8^2\)[/tex], we need to calculate the squared distance of each point from the center of the circle and compare it to the square of the circle's radius.
The equation of the circle is [tex]\((x + 5)^2 + (y - 9)^2 = 8^2\)[/tex].
First, identify the center and radius of the circle:
- Center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-5, 9)\)[/tex]
- Radius [tex]\(r\)[/tex] is [tex]\(8\)[/tex]
### Check each point:
1. Point (0, 8):
[tex]\[
(0 + 5)^2 + (8 - 9)^2 = 5^2 + (-1)^2 = 25 + 1 = 26
\][/tex]
2. Point (13, -9):
[tex]\[
(13 + 5)^2 + (-9 - 9)^2 = 18^2 + (-18)^2 = 324 + 324 = 648
\][/tex]
3. Point (-5, 1):
[tex]\[
(-5 + 5)^2 + (1 - 9)^2 = 0^2 + (-8)^2 = 0 + 64 = 64
\][/tex]
4. Point (3, 17):
[tex]\[
(3 + 5)^2 + (17 - 9)^2 = 8^2 + 8^2 = 64 + 64 = 128
\][/tex]
The squared radius of the circle is:
[tex]\[ 8^2 = 64 \][/tex]
Now let's compare each calculated squared distance with the squared radius:
- For [tex]\((0, 8)\)[/tex], the squared distance is [tex]\(26 \neq 64\)[/tex].
- For [tex]\((13, -9)\)[/tex], the squared distance is [tex]\(648 \neq 64\)[/tex].
- For [tex]\((-5, 1)\)[/tex], the squared distance is [tex]\(64 = 64\)[/tex].
- For [tex]\((3, 17)\)[/tex], the squared distance is [tex]\(128 \neq 64\)[/tex].
Only the point [tex]\((-5, 1)\)[/tex] satisfies the equation of the circle.
Therefore, the correct answer is:
C. [tex]\((-5, 1)\)[/tex]
