Answer :
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]
### 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]
