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 if each given point satisfies this equation. The equation [tex]\((x-3)^2+(y+4)^2=6^2\)[/tex] represents a circle with center at [tex]\((3, -4)\)[/tex] and a radius of [tex]\(6\)[/tex].
Let's check each point:
### Check Point [tex]\((9, -2)\)[/tex]:
1. Substitute [tex]\(x = 9\)[/tex] and [tex]\(y = -2\)[/tex] into the circle equation:
2. [tex]\((9-3)^2 + (-2+4)^2 = 6^2\)[/tex]
3. [tex]\(6^2 + 2^2 = 36\)[/tex]
4. [tex]\(36 + 4 = 40\)[/tex]
5. [tex]\(40 \neq 36\)[/tex]
So, [tex]\((9, -2)\)[/tex] does not satisfy the equation.
### Check Point [tex]\((0, 11)\)[/tex]:
1. Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 11\)[/tex] into the circle equation:
2. [tex]\((0-3)^2 + (11+4)^2 = 6^2\)[/tex]
3. [tex]\((-3)^2 + 15^2 = 36\)[/tex]
4. [tex]\(9 + 225 = 234\)[/tex]
5. [tex]\(234 \neq 36\)[/tex]
So, [tex]\((0, 11)\)[/tex] does not satisfy the equation.
### Check Point [tex]\((3, 10)\)[/tex]:
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 10\)[/tex] into the circle equation:
2. [tex]\((3-3)^2 + (10+4)^2 = 6^2\)[/tex]
3. [tex]\(0^2 + 14^2 = 36\)[/tex]
4. [tex]\(0 + 196 = 196\)[/tex]
5. [tex]\(196 \neq 36\)[/tex]
So, [tex]\((3, 10)\)[/tex] does not satisfy the equation.
### Check Point [tex]\((-9, 4)\)[/tex]:
1. Substitute [tex]\(x = -9\)[/tex] and [tex]\(y = 4\)[/tex] into the circle equation:
2. [tex]\((-9-3)^2 + (4+4)^2 = 6^2\)[/tex]
3. [tex]\((-12)^2 + 8^2 = 36\)[/tex]
4. [tex]\(144 + 64 = 208\)[/tex]
5. [tex]\(208 \neq 36\)[/tex]
So, [tex]\((-9, 4)\)[/tex] does not satisfy the equation.
### Check Point [tex]\((-3, -4)\)[/tex]:
1. Substitute [tex]\(x = -3\)[/tex] and [tex]\(y = -4\)[/tex] into the circle equation:
2. [tex]\((-3-3)^2 + (-4+4)^2 = 6^2\)[/tex]
3. [tex]\((-6)^2 + 0^2 = 36\)[/tex]
4. [tex]\(36 + 0 = 36\)[/tex]
5. [tex]\(36 = 36\)[/tex]
So, [tex]\((-3, -4)\)[/tex] does satisfy the equation.
Hence, the point [tex]\((-3, -4)\)[/tex] lies on the circle. Therefore, the correct answer is:
E. [tex]\((-3, -4)\)[/tex]
