Three Dimensional Geometry

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

The shortest distance, between lines L1 and L2, where L1: x-12= y+1-3= z+42 and L2 is the line, passing through the points A(-4,4,3), B(-1,6,3) and perpendicular to the line x-3-2= y3= z-11, is

1. 141 221
2. 24 117
3. 42 117
4. 121 221

A line passes through A(4,-6,-2) and B(16,-2,4). The point P(a, b, c), where a, b, c are non-negative integers, on the line AB lies at a distance of 21 units, from the point A. The distance between the points P(a, b, c) and Q(4,-12,3) is equal to .

Let ( , , ) be the foot of perpendicular from the point (1,2,3) on the line x+35= y-12= z+43. Then 19( + + ) is equal to :

1. 99
2. 102
3. 101
4. 100

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 :