Three Dimensional Geometry

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 :

Let L1: r=( i- j+2 k)+ ( i- j+2 k), R, L2: r=( j- k)+ (3 i+ j+p k), R, and L3: r= ( i+m j+n k), R be three lines such that L1 is perpendicular to L2 and L3 is perpendicular to both L1 and L2. Then, the point which lies on L3 is

1. (1,7,-4)
2. (1,-7,4)
3. (-1,7,4)
4. (-1,-7,4)

Let a line passing through the point (-1,2,3) intersect the lines L1: x-13= y-22= z+1-2 at M( , , ) and L2: x+2-3= y-2-2= z-14 at N(a, b, c). Then, the value of ( + + )2(a+b+c)2 equals .

Let PQR be a triangle with R(-1,4,2). Suppose M(2,1,2) is the mid point of PQ. The distance of the centroid of from the point of intersection of the lines x-20= y2= z+3-1 and x-11= y+3-3= z+11 is

1. 69
2. 99
3. 69
4. 9

A line with direction ratios 2,1,2 meets the lines x=y+2=z and x+2=2y=2z respectively at the points P and Q. If the length of the perpendicular from the point (1,2,12) to the line PQ is l, then l2 is .