Let's analyze the given options to determine which point lies on the circle defined by the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex].
First, we note that the center of the circle is [tex]\((3, -4)\)[/tex] and its radius is [tex]\(6\)[/tex].
We need to check each given point to see if it satisfies the circle's equation.
Option 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]\[
36 + 4 = 40 \neq 36
\][/tex]
So, [tex]\((9, -2)\)[/tex] does not lie on the circle.
Option 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]\[
9 + 225 = 234 \neq 36
\][/tex]
So, [tex]\((0, 11)\)[/tex] does not lie on the circle.
Option 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]\[
0 + 196 = 196 \neq 36
\][/tex]
So, [tex]\((3, 10)\)[/tex] does not lie on the circle.
Option 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]\[
144 + 64 = 208 \neq 36
\][/tex]
So, [tex]\((-9, 4)\)[/tex] does not lie on the circle.
Option 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]\[
36 + 0 = 36
\][/tex]
So, [tex]\((-3, -4)\)[/tex] does lie on the circle.
Thus, the point [tex]\((-3, -4)\)[/tex] is the correct answer. Therefore, the correct option is:
E. [tex]\((-3, -4)\)[/tex]
