To determine which point lies on the circle represented by the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], we need to check each given point using this equation. The equation represents a circle centered at [tex]\((3, -4)\)[/tex] with a radius of 6.
Let's test each point step-by-step:
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]\[
36 + 4 = 40 \quad (\text{not equal to } 36)
\][/tex]
Hence, (9, -2) 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]\[
9 + 225 = 234 \quad (\text{not equal to } 36)
\][/tex]
Hence, (0, 11) 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]\[
0 + 196 = 196 \quad (\text{not equal to } 36)
\][/tex]
Hence, (3, 10) 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]\[
144 + 64 = 208 \quad (\text{not equal to } 36)
\][/tex]
Hence, (-9, 4) 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]\[
36 + 0 = 36 \quad (\text{equal to } 36)
\][/tex]
Hence, (-3, -4) lies on the circle.
Thus, the correct answer is:
E. [tex]\((-3, -4)\)[/tex]
