If the shortest distance between the lines x- k 2 = y -4 3 = z -3 4 and x-2 4 = y -4 6 = z -7 8 is 13 29 , then k =

Step-by-step solution

The line L₁: x-k 2 = y-4 3 = z-3 4 passes through A(k, 4, 3) with direction vector d₁ = 2 i + 3 j + 4 k . The line L₂: x-2 4 = y-4 6 = z-7 8 passes through B(2, 4, 7) with direction vector d₂ = 4 i + 6 j + 8 k . Observe that d₂ = 2 d₁ , so the lines are parallel. The vector connecting points A and B is AB = (2-k) i + 0 j + 4 k . The shortest distance between parallel lines is given by D = | AB d₁ | | d₁ | where | d₁ | = 29 . The cross product yields AB d₁ = -12 i + 4k j + (6-3k) k , with magnitude | AB d₁ | = 25k^2