To determine which point lies on the circle represented by the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], we will check each given point to see if it satisfies the equation of the circle.
The equation of the circle is:
[tex]\[
(x - 3)^2 + (y + 4)^2 = 36
\][/tex]
We will evaluate each point:
1. Point A: [tex]\((9, -2)\)[/tex]
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]\[
36 + 4 = 40 \quad \text{(not 36, so not on the circle)}
\][/tex]
2. Point B: [tex]\((0, 11)\)[/tex]
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]\[
9 + 225 = 234 \quad \text{(not 36, so not on the circle)}
\][/tex]
3. Point C: [tex]\((3, 10)\)[/tex]
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]\[
0 + 196 = 196 \quad \text{(not 36, so not on the circle)}
\][/tex]
4. Point D: [tex]\((-9, 4)\)[/tex]
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]\[
144 + 64 = 208 \quad \text{(not 36, so not on the circle)}
\][/tex]
5. Point E: [tex]\((-3, -4)\)[/tex]
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]\[
36 + 0 = 36 \quad \text{(which equals 36, so it is on the circle)}
\][/tex]
Therefore, the correct answer is:
E. [tex]\((-3, -4)\)[/tex]
