Three Dimensional Geometry

If the shortest distance between the lines x - 3 = y - 2-1 = z - 11 and x + 2-3 = y + 52 = z - 44 is 44 30, then the largest possible value of | | is equal to .

Let the point (-1, , ) lie on the line of the shortest distance between the lines x + 2-3 = y - 24 = z - 52 and x + 2-1 = y + 62 = z - 10. Then ( - )2 is equal to .

If the shortest distance between the lines x+22= y+33= z-54 and x-31= y-2-3= z+42 is 383 5k, and 0k [x2 ]dx= - , where [x] denotes the greatest integer function, then 6 3 is equal to .

Let the line of the shortest distance between the lines L1: r=( i+2 j+3 k)+ ( i- j+ k) and L2: r=(4 i+5 j+6 k)+ ( i+ j- k) intersect L1 and L2 at P and Q respectively. If ( , , ) is the mid point of the line segment PQ, then 2( + + ) is equal to .

If d1 is the shortest distance between the lines x+1=2y=-12z, x=y+2=6z-6 and d2 is the shortest distance between the lines x-12= y+8-7= z-45, x-12= y-21= z-6-3, then the value of 32 3 d1d2 is :

Let O be the origin, and M and N be the points on the lines x-54= y-41= z-53 and x+812= y+25= z+119 respectively such that MN is the shortest distance between the given lines. Then OM ON is equal to .