Three Dimensional Geometry

Let the foot of perpendicular from the point ( , 2, 3) on the line x-41= y-92= z-51 be the point (1, , 2). Then the distance between the lines x-12= y-23= z+46 and x- 2= y- 3= z+56 is equal to :

1. 127
2. 1457
3. 1467
4. 1437

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

The square of the distance of the point (-2, -8, 6) from the line x-11= y-12= z-1 along the line x+51= y+5-1= z2 is equal to:

1. 3
2. 6
3. 8
4. 12

The shortest distance between the lines r= ( 13 i+2 j+ 83 k )+ (2 i-5 j+6 k ) and r= (- 23 i- 13 k )+ ( j- k ), , R, is :

1. 5
2. 3
3. 2 3
4. 15

If (2 +1, 2-3 , -12 ) is the image of ( , 2 , 1) in the line x-23= y-12= z1, then the possible value(s) of is (are)

1. Only 3
2. Only 3 and -1
3. Only 3, 14 and -1
4. Only 3 and 14

If the point of intersection of the lines x+13= y+a5= z+b+17 and x-21= y-b4= z-2a7 lies on xy-plane, then the value of a+b is :

1. 2
2. 5
3. 7
4. 9

Let a line L passing through the point (1, 1, 1) be perpendicular to both the vectors 2 i+2 j+ k and i+2 j+2 k. If P(a, b, c) is the foot of perpendicular from the origin on the line L, then the value of 34(a+b+c) is :

1. 50
2. 80
3. 100
4. 120

Let the point A be the foot of perpendicular drawn from the point P(a, b, 0) on the line x-12= y-21= z- 3. If the midpoint of the line segment PA is (0, 34, - 14 ), then the value of a2+b2+ 2 is equal to :

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

If the distances of the point (1, 2, a) from the line x - 11 = y2 = z - 11 along the lines L1 : x - 13 = y - 24 = z - ab and L2 : x - 11 = y - 24 = z - ac are equal, then a + b + c is equal to

1. 4
2. 6
3. 7
4. 5

Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line x - 11 = y2 = z - 11. Then the distance of Q from the line x - 93 = y - 92 = z - 5-2 is

1. 8
2. 7
3. 6
4. 5

If the distance of the point P(43, , ), < 0, from the line r = 4 i - k + (2 i + 3 k), R along a line with direction ratios 3, -1, 0 is 13 10, then 2 + 2 is equal to

Let the lines L1: r = i + 2 j + 3 k + (2 i + 3 j + 4 k), R and L2: r = (4 i + j) + (5 i + 2 j + k), R, intersect at the point R. Let P and Q be the points lying on lines L1 and L2, respectively, such that | PR| = 29 and | PQ| = 473. If the point P lies in the first octant, then 27(QR)2 is equal to

1. 360
2. 348
3. 320
4. 340

Let a line L passing through the point P(1, 1, 1) be perpendicular to the lines x-44 = y-11 = z-11 and x-171 = y-711 = z0. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point S(1, 0, -1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to .

The sum of all values of , for which the shortest distance between the lines x+1 = y-2-1 = z-4- and x = y-12 = z-12 is 2, is

1. -6
2. -8
3. 8
4. 6

Let the direction cosines of two lines satisfy the equations : 4l + m - n = 0 and 2mn + 10nl + 3lm = 0. Then the cosine of the acute angle between these lines is :

1. 107 38
2. 10 38
3. 103 38
4. 203 38

The vertices B and C of a triangle ABC lie on the line x1 = 1 - y-2 = z - 23. The coordinates of A and B are (1, 6, 3) and (4, 9, α) respectively and C is at a distance of 10 units from B. The area (in sq. units) of ABC is :

1. 20 13
2. 5 13
3. 15 13
4. 10 13

If the image of the point P(a, 2, a) in the line x2 = y + a1 = z1 is Q and the image of Q in the line x - 2b2 = y - a1 = z + 2b-5 is P , then a + b is equal to .

Let P( , , ) be the point on the line x - 12 = y + 1-3 = z at a distance 4 14 from the point (1, -1, 0) and nearer to the origin. Then the shortest distance, between the lines x - 1 = y - 2 = z - 3 and x + 52 = y - 101 = z - 31, is equal to

1. 4 75
2. 7 54
3. 4 57
4. 2 74

If the image of the point P(1, 2, a) in the line x - 63 = y - 72 = 7 - z2 is Q(5, b, c), then a2 + b2 + c2 is equal to

1. 298
2. 264
3. 293
4. 283

Let L be the line x+12 = y+13 = z+36 and let S be the set of all points (a, b, c) on L, whose distance from the line x+12 = y+13 = z-90 along the line L is 7. Then (a, b, c) S (a + b + c) is equal to :

1. 28
2. 6
3. 40
4. 34

Let ( , , ) be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line r = (- i + 3 j + k) + (2 i + 3 j - k). Then the length of the projection of the vector i + j + k on the vector 6 i + 2 j + 3 k is :

1. 187
2. 157
3. 4
4. 3