If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line x 1 = y-1 1 = z-2 2 is 4 , then the sum of all possible values of a is equal to :

Step-by-step solution

Let P(1, 2, 7) be the given point and the given line be L: x 1 = y-1 1 = z-2 2 = . Any point on the line can be written as F( , +1, 2 +2) . If F is the foot of the perpendicular from P to the line, the direction ratios of PF are ( -1, -1, 2 -5) . Since PF is perpendicular to the line, the dot product of their direction ratios is zero: 1( -1) + 1( -1) + 2(2 -5) = 0 6 - 12 = 0 = 2 Thus, the foot of the perpendicular is F(2, 3, 6) . Let the image of P in the line be P'(x', y', z') . Since F is the midpoint of P and P'