Let ( α , β , γ ) be the foot of perpendicular from the point ( 1 , 2 , 3 ) on the line x + 3 5 = y - 1 2 = z + 4 3 . then 19 ( α + β + γ ) is equal to :

Step-by-step solution

Given: P α , β , γ be the foot of the perpendicular from the point Q 1 , 2 , 3 on the line x + 3 5 = y - 1 2 = z + 4 3 Now, let x + 3 5 = y - 1 2 = z + 4 3 = λ So, any point on the line is given by 5 λ - 3 , 2 λ + 1 , 3 λ - 4 , so let the point be P 5 λ - 3 , 2 λ + 1 , 3 λ - 4 Now, direction ratio of P Q = 5 λ - 4 , 2 λ - 1 , 3 λ - 7 Now, the line and P Q will be perpendicular, So, 5 5 λ - 4 + 2 2 λ - 1 + 3 3 λ - 7 = 0 ⇒ 38 λ - 43 = 0 ⇒ λ = 43 38 So, the point P α , β , γ = P 5 × 43 38 - 3 , 2 × 43 38 + 1 , 3 × 43