Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?

Let the equation of the parabola be y² = 4ax, where a is the distance between the vertex and the focus. The vertex is at (2, 0) and the focus is at (4, 0). Therefore, the distance between the vertex and focus is 4 - 2 = 2. So, a = 2. The equation of the parabola is y² = 4(2)(x - 2) which simplifies to y² = 8(x - 2). Now let's check which point does not satisfy this equation:

(5, 2√6): (2√6)² = 8(5 - 2) => 24 = 24. This point lies on the parabola.
(8, 6): 6² = 8(8 - 2) => 36 = 48. This point does not lie on the parabola.
(4, √8): (√8)² = 8(4 - 2) => 8 = 16. This point does not lie on the parabola.
(6, 4√2): (4√2)² = 8(6 - 2) => 32 = 32. This point lies on the parabola.

Therefore, (8, 6) and (4, √8) do not lie on the parabola.