The equation given, (x - 3)² + (y + 4)² = 6², represents a circle with its center at the point (3, -4) and a radius of 6 units. To determine which point lies on this circle, we need to check which point satisfies the equation of the circle.
Let's substitute the points given in the options into the equation and see which one satisfies it:
A. (9, -2)
Plugging in (x, y) = (9, -2) into the equation:
(9 - 3)² + (-2 + 4)² = 6²
(6)² + (2)² = 6²
36 + 4 = 36
This point does not lie on the circle as the equation is not satisfied.
B. (0, 11)
Plugging in (x, y) = (0, 11) into the equation:
(0 - 3)² + (11 + 4)² = 6²
(-3)² + (15)² = 6²
9 + 225 = 36
This point does not lie on the circle as the equation is not satisfied.
C. (3, 10)
Plugging in (x, y) = (3, 10) into the equation:
(3 - 3)² + (10 + 4)² = 6²
(0)² + (14)² = 6²
0 + 196 = 36
This point does not lie on the circle as the equation is not satisfied.
D. (-9, 4)
Plugging in (x, y) = (-9, 4) into the equation:
(-9 - 3)² + (4 + 4)² = 6²
(-12)² + (8)² = 6²
144 + 64 = 36
This point does not lie on the circle as the equation is not satisfied.
E. (-3, -4)
Plugging in (x, y) = (-3, -4) into the equation:
(-3 - 3)² + (-4 + 4)² = 6²
(-6)² + (0)² = 6²
36 + 0 = 36
This point lies on the circle as the equation is satisfied.
Therefore, the point that lies on the circle represented by the equation is (-3, -4) (Option E).
