If the shortest distance between the line x - 1 α = y + 1 - 1 = z 1 , α ≠ - 1 , and x + y + z + 1 = 0 = 2 x - y + z + 3 is 1 3 ,then value of α is :

Step-by-step solution

Let us change the line into symmetric form. x + y + z + 1 = 0 = 2 x - y + z + 3 Put z = 1 , so we get x + y + 2 = 0 and 2 x - y + 4 = 0  We will get x = - 2 , y = 0 ∴ The point - 2 , 0 , 1 lies on the line and perpendicular vector will come from i ^ j ^ k ^ 1 1 1 2 - 1 1 = 2 i ^ + j ^ - 3 k ^ So the equation of line would be x + 2 2 = y 1 = z - 1 - 3 And the other line x - 1 α = y + 1 - 1 = z 1 Shortest distance would be D = a 2 → - a 1 → · b 1 → × b 2 → b 1 →