The equation of the perpendicular drawn from the point A(6, 1, 3) to the line x-1 2 = 2-y -1 = z-3 2 is x-6 a = y-1 b = z-3 c . If a, b, c are the possible integers such that a < 0 , then the value of a - b + 5c is:

Step-by-step solution

The given line can be rewritten in standard form as: x-1 2 = y-2 1 = z-3 2 Let the foot of the perpendicular from A(6, 1, 3) to the given line be P . Any point on the line can be taken as P(2 + 1, + 2, 2 + 3) . The direction ratios of the line segment AP are: (2 + 1 - 6, + 2 - 1, 2 + 3 - 3) = (2 - 5, + 1, 2 ) Since AP is perpendicular to the given line, the dot product of their direction ratios is zero: 2(2 - 5) + 1( + 1) + 2(2 ) = 0 4 - 10 + + 1 + 4 = 0 9 - 9 = 0 = 1 Substituting = 1 , the coordinates of P are (3,