Three Dimensional Geometry

Let a line L1 pass through the origin and be perpendicular to the lines L2 : r=(3+t) i+(2t-1) j+(2t+4) k and L3 : r=(3+2s) i+(3+2s) j+(2+s) k, t, s R. If (a, b, c), a Z, is the point on L3 at a distance of 17 from the point of intersection of L1 and L2, then (a+b+c)2 is equal to .

Let the image of the point P(1, 6, a) in the line L : x1= y-12= z-a+1b, b>0, be ( a3, 0, a+c ). If S( , , ), >0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2 14, then + + is equal to :

1. 19
2. 20
3. 21
4. 22

Let a line L be perpendicular to both the lines L1 : x+13= y+35= z+57 and L2 : x-21= y-44= z-67. If is the acute angle between the lines L and L3 : x- 872= y- 471= z2, then is equal to :

1. 32 2
2. 52 2
3. 53 2
4. 43 2

The shortest distance between the lines x-41= y-32= z-2-3 and x+22= y-64= z-5-5 is :

1. 5 66
2. 2 5
3. 3 5
4. 4 5

Let the image of the point P(0, -5, 0) in the line x-12= y1= z+1-2 be the point R and the image of the point Q (0, -12, 0 ) in the line x-1-1= y+94= z+11 be the point S. Then the square of the area of the parallelogram PQRS is .

The square of the distance of the point P(5, 6, 7) from the line x-22= y-53= z-24 is equal to:

1. 3
2. 5
3. 6
4. 8

The square of the distance of the point of intersection of the lines r= ( i+ j- k )+ (a i- j ), a 0 and r= (4 i- k )+ (2 i+a k ) from the origin is :

1. 5
2. 10
3. 17
4. 26

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

1. 4
2. 5
3. 6
4. 8

If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line x1= y-11= z-22 is 4, then the sum of all possible values of a is equal to :

1. 11
2. 9
3. 6
4. 4

A line with direction ratios 1, -1, 2 intersects the lines x2= y3= z+13 and x+1-1= y-21= z4 at the points P and Q, respectively. If the length of the line segment PQ is , then 225 2 is equal to:

1. 1024
2. 1014
3. 1104
4. 1204