Three Dimensional Geometry

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 P be the point of intersection of the lines x - 21 = y - 45 = z - 21 and x - 32 = y - 23 = z - 32. Then, the shortest distance of P from the line 4x = 2y = z is

1. 3 147
2. 5 147
3. 147
4. 6 147

Consider a line L passing through the points P(1, 2, 1) and Q(2, 1, -1). If the mirror image of the point A(2, 2, 2) in the line L is ( , , ), then + + 6 is equal to .

If the shortest distance between the lines x- -2= y-21= z-11 and x- 31= y-1-2= z-21 is 1 , then the sum of all possible values of is :

1. 0
2. 2 3
3. 3 3
4. -2 3

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 .

Let P and Q be the points on the line x+38= y-42= z+12 which are at a distance of 6 units from the point R(1,2,3). If the centroid of the triangle PQR is ( , , ), then 2+ 2+ 2 is :

1. 18
2. 24
3. 26
4. 36

If the mirror image of the point P(3,4,9) in the line x-13= y+12= z-21 is ( , , ), then 14( + + ) is :

1. 102
2. 138
3. 132
4. 108

The distance of the point Q(0,2,-2) from the line passing through the point P(5,-4,3) and perpendicular to the lines r=(-3 i+2 k)+ (2 i+3 j+5 k), R and r=( i-2 j+ k)+ (- i+3 j+2 k), R is :

1. 74
2. 86
3. 54
4. 20

Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x=y, z=1 and x=-y, z=-1 respectively. If is a right angle, then 12a2 is equal to .

Let ( , , ) be the mirror image of the point (2,3,5) in the line x-12= y-23= z-34. Then, 2 +3 +4 is equal to

1. 32
2. 31
3. 33
4. 34