The line of shortest distance between the lines x - 2 0 = y - 1 1 = z 1 and x - 3 2 = y - 5 2 = z - 1 1 makes an angle of sin - 1 2 27 with the plane P : a x - y - z = 0 , a > 0 . If the image of the point 1 , 1 , - 5 in the plane P is α , β , γ , then α + β - γ is equal to _____ .

Step-by-step solution

Given lines, x - 2 0 = y - 1 1 = z 1 and x - 3 2 = y - 5 2 = z - 1 1 Now direction ratio of line of shortest distance will be, i ^ j ^ k ^ 0 1 1 2 2 1 = - i ^ + 2 j ^ - 2 k ^ Now given angle between line and plane is sin - 1 2 27 = α , so sin α = 2 27 Now direction ratio of normal to plane is a , - 1 , - 1 So, sin α = - a - 2 + 2 4 + 4 + 1 a 2 + 1 + 1 = 2 3 3 ⇒ 3 a = 2 a 2 + 2 ⇒ 3 a 2 = 2 a 2 + 4 ⇒ a 2 = 4 ⇒ a = 2   as   a > 0   given Now equation of plane w