Let's determine which point lies on the circle represented by the equation [tex]\((x - 3)^2 + (y + 4)^2 = 6^2\)[/tex].
We know that the equation of the circle is given by:
[tex]\[
(x - 3)^2 + (y + 4)^2 = 36
\][/tex]
To find which point lies on the circle, we need to check each given point to see if it satisfies the equation.
1. Point A: [tex]\((9, -2)\)[/tex]
Substitute [tex]\( x = 9 \)[/tex] and [tex]\( y = -2 \)[/tex] into the circle equation:
[tex]\[
(9 - 3)^2 + (-2 + 4)^2 = 6^2
\][/tex]
[tex]\[
(6)^2 + (2)^2 = 36
\][/tex]
[tex]\[
36 + 4 = 40 \quad \text{(not on the circle)}
\][/tex]
2. Point B: [tex]\((0, 11)\)[/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 11 \)[/tex] into the circle equation:
[tex]\[
(0 - 3)^2 + (11 + 4)^2 = 6^2
\][/tex]
[tex]\[
(-3)^2 + (15)^2 = 36
\][/tex]
[tex]\[
9 + 225 = 234 \quad \text{(not on the circle)}
\][/tex]
3. Point C: [tex]\((3, 10)\)[/tex]
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 10 \)[/tex] into the circle equation:
[tex]\[
(3 - 3)^2 + (10 + 4)^2 = 6^2
\][/tex]
[tex]\[
(0)^2 + (14)^2 = 36
\][/tex]
[tex]\[
0 + 196 = 196 \quad \text{(not on the circle)}
\][/tex]
4. Point D: [tex]\((-9, 4)\)[/tex]
Substitute [tex]\( x = -9 \)[/tex] and [tex]\( y = 4 \)[/tex] into the circle equation:
[tex]\[
(-9 - 3)^2 + (4 + 4)^2 = 6^2
\][/tex]
[tex]\[
(-12)^2 + (8)^2 = 36
\][/tex]
[tex]\[
144 + 64 = 208 \quad \text{(not on the circle)}
\][/tex]
5. Point E: [tex]\((-3, -4)\)[/tex]
Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = -4 \)[/tex] into the circle equation:
[tex]\[
(-3 - 3)^2 + (-4 + 4)^2 = 6^2
\][/tex]
[tex]\[
(-6)^2 + (0)^2 = 36
\][/tex]
[tex]\[
36 + 0 = 36 \quad \text{(on the circle)}
\][/tex]
Therefore, the point that lies on the circle is [tex]\((-3, -4)\)[/tex].
The answer is:
[tex]\[ \boxed{E} \][/tex]
