If the vertices of a hyperbola are at (-6,0) and (2,0) and one of its foci is at (7,0), then which one of the following points does not lie on this hyperbola?

Let the vertices of the hyperbola be A(-6, 0) and B(2, 0). The center of the hyperbola is the midpoint of AB, which is ((-6+2)/2, (0+0)/2) = (-2, 0).
Let the focus be F(7, 0). The distance between the center and a vertex is a = 2 - (-2) = 4. The distance between the center and a focus is c = 7 - (-2) = 9.
The relationship between a, b, and c in a hyperbola is c² = a² + b². Therefore, b² = c² - a² = 9² - 4² = 81 - 16 = 65.
The equation of the hyperbola with center (-2, 0) is:
(x + 2)²/a² - y²/b² = 1
(x + 2)²/16 - y²/65 = 1
Now let's check each point:

1. (2√2, √5):
   ((2√2) + 2)²/16 - (√5)²/65 = (2√2 + 2)²/16 - 5/65 ≈ 0.88 - 0.077 ≈ 0.8 ≠ 1
2. (4, √15):
   (4 + 2)²/16 - (√15)²/65 = 36/16 - 15/65 = 9/4 - 3/13 ≈ 2.25 - 0.23 ≈ 2 ≠ 1
3. (-8, 2√10):
   (-8 + 2)²/16 - (2√10)²/65 = 36/16 - 40/65 = 9/4 - 8/13 ≈ 2.25 - 0.615 ≈ 1.63 ≠ 1
4. (6, 5√2):
   (6 + 2)²/16 - (5√2)²/65 = 64/16 - 50/65 = 4 - 10/13 ≈ 4 - 0.77 ≈ 3.23 ≠ 1
   Based on these calculations, none of the points perfectly satisfy the equation of the hyperbola. There might be a calculation error in the original solution or the provided options. However, the closest point to satisfying the equation is (2√2, √5).