If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x - y = 15, then 2α - β is equal to:

Let the given points be A(3, 4, 2) and B(7, 0, 6).
The direction ratios of AB are (7-3, 0-4, 6-2) = (4, -4, 4).
The plane passes through A(3, 4, 2) and is perpendicular to the plane 2x - y = 15.
The normal vector to the plane 2x - y = 15 is (2, -1, 0).
Let the equation of the plane be 2x - y + c = 0.
Since the plane passes through A(3, 4, 2), we have:
2(3) - 4 + c = 0
6 - 4 + c = 0
c = -2
Therefore, the equation of the plane is 2x - y - 2 = 0.
The point (2, α, β) lies on this plane, so:
2(2) - α - 2 = 0
4 - α - 2 = 0
α = 2
The plane also passes through A(3, 4, 2) and B(7, 0, 6).
Let the equation of the plane be ax + by + cz = d.
Then:
3a + 4b + 2c = d
7a + 6c = d
The direction ratios of the normal to this plane are (a, b, c).
Since the plane is perpendicular to 2x - y = 15, the dot product of their normal vectors is 0:
2a - b = 0
b = 2a
Let the equation of the plane be ax + 2ay + cz = d.
Then:
3a + 8a + 2c = d
11a + 2c = d
7a + 6c = d
11a + 2c = 7a + 6c
4a = 4c
a = c
So the equation of the plane is ax + 2ay + az = d.
Since (2, α, β) lies on the plane:
2a + 2αa + βa = d
Since (3, 4, 2) lies on the plane:
3a + 8a + 2a = d
13a = d
Since (7, 0, 6) lies on the plane:
7a + 6a = d
13a = d
Thus, 2a + 2αa + βa = 13a
2 + 2α + β = 13
2α + β = 11
We have α = 2.
Then 2(2) + β = 11
β = 7
2α - β = 2(2) - 7 = 4 - 7 = -3
However, this contradicts the given options. Let's re-examine the solution.
The vector perpendicular to the plane 2x - y = 15 is (2, -1, 0).
Let the plane be 2x - y + c = 0. Since (3,4,2) is on the plane, 2(3) - 4 + c = 0, so c = -2.
The equation of the plane is 2x - y - 2 = 0.
Since (2, α, β) lies on this plane, 2(2) - α - 2 = 0, which gives α = 2.
The vector AB is (4, -4, 4). The plane containing AB and normal to (2, -1, 0) has normal vector (4, -4, 4) x (2, -1, 0) = (4, 8, 4) = 4(1, 2, 1).
The equation of this plane is x + 2y + z = d. Since (3, 4, 2) is on it, 3 + 8 + 2 = 13 = d.
So x + 2y + z = 13. Since (2, α, β) is on it, 2 + 2α + β = 13, so 2α + β = 11. Since α = 2, β = 7.
2α - β = 2(2) - 7 = -3. There must be a mistake in the problem statement or options.