To determine which point lies on the circle represented by the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], we can follow these steps:
1. Understand the Equation: A circle with a center at [tex]\((3, -4)\)[/tex] and radius [tex]\(6\)[/tex] is given by the equation [tex]\((x-3)^2 + (y+4)^2 = 36\)[/tex].
2. Plugging the Points into the Circle's Equation:
- Option A: [tex]\((9, -2)\)[/tex]
[tex]\[
(9-3)^2 + (-2+4)^2 = 6^2
\][/tex]
[tex]\[
6^2 + 2^2 = 36
\][/tex]
[tex]\[
36 + 4 = 40 \quad (\text{not equal to } 36)
\][/tex]
Therefore, [tex]\((9, -2)\)[/tex] is not on the circle.
- Option B: [tex]\((0, 11)\)[/tex]
[tex]\[
(0-3)^2 + (11+4)^2 = 6^2
\][/tex]
[tex]\[
(-3)^2 + 15^2 = 36
\][/tex]
[tex]\[
9 + 225 = 234 \quad (\text{not equal to } 36)
\][/tex]
Therefore, [tex]\((0, 11)\)[/tex] is not on the circle.
- Option C: [tex]\((3, 10)\)[/tex]
[tex]\[
(3-3)^2 + (10+4)^2 = 6^2
\][/tex]
[tex]\[
0^2 + 14^2 = 36
\][/tex]
[tex]\[
0 + 196 = 196 \quad (\text{not equal to } 36)
\][/tex]
Therefore, [tex]\((3, 10)\)[/tex] is not on the circle.
- Option D: [tex]\(( -9, 4)\)[/tex]
[tex]\[
(-9-3)^2 + (4+4)^2 = 6^2
\][/tex]
[tex]\[
(-12)^2 + 8^2 = 36
\][/tex]
[tex]\[
144 + 64 = 208 \quad (\text{not equal to } 36)
\][/tex]
Therefore, [tex]\((-9, 4)\)[/tex] is not on the circle.
- Option E: [tex]\(( -3, -4)\)[/tex]
[tex]\[
(-3-3)^2 + (-4+4)^2 = 6^2
\][/tex]
[tex]\[
(-6)^2 + 0^2 = 36
\][/tex]
[tex]\[
36 + 0 = 36 \quad (\text{equal to } 36)
\][/tex]
Therefore, [tex]\((-3, -4)\)[/tex] is on the circle.
After evaluating each point, we find that the correct answer is:
E. [tex]\((-3, -4)\)[/tex]
