To determine which of the given points lies on the circle represented by the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], we'll substitute each point into the circle's equation and see which one satisfies it.
1. Point (9, -2)
- Substitute [tex]\( x = 9 \)[/tex] and [tex]\( y = -2 \)[/tex] into the equation:
[tex]\[
(9-3)^2 + (-2+4)^2 = 6^2
\][/tex]
[tex]\[
(6)^2 + (2)^2 = 36
\][/tex]
[tex]\[
36 + 4 = 40 \neq 36
\][/tex]
This point does not lie on the circle.
2. Point (0, 11)
- Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 11 \)[/tex] into the equation:
[tex]\[
(0-3)^2 + (11+4)^2 = 6^2
\][/tex]
[tex]\[
(-3)^2 + (15)^2 = 36
\][/tex]
[tex]\[
9 + 225 = 234 \neq 36
\][/tex]
This point does not lie on the circle.
3. Point (3, 10)
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 10 \)[/tex] into the equation:
[tex]\[
(3-3)^2 + (10+4)^2 = 6^2
\][/tex]
[tex]\[
(0)^2 + (14)^2 = 36
\][/tex]
[tex]\[
0 + 196 = 196 \neq 36
\][/tex]
This point does not lie on the circle.
4. Point (-9, 4)
- Substitute [tex]\( x = -9 \)[/tex] and [tex]\( y = 4 \)[/tex] into the equation:
[tex]\[
(-9-3)^2 + (4+4)^2 = 6^2
\][/tex]
[tex]\[
(-12)^2 + (8)^2 = 36
\][/tex]
[tex]\[
144 + 64 = 208 \neq 36
\][/tex]
This point does not lie on the circle.
5. Point (-3, -4)
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = -4 \)[/tex] into the equation:
[tex]\[
(-3-3)^2 + (-4+4)^2 = 6^2
\][/tex]
[tex]\[
(-6)^2 + (0)^2 = 36
\][/tex]
[tex]\[
36 + 0 = 36 = 36
\][/tex]
This point satisfies the equation of the circle.
Therefore, the point that lies on the circle is [tex]\( \boxed{(-3, -4)} \)[/tex].
