Three Dimensional Geometry

The distance of the line x-22= y-63= z-34 from the point (1,4,0) along the line x1= y-22= z+33 is :

1. 17
2. 13
3. 15
4. 14

Let L1: x-12= y-23= z-34 and L2: x-23= y-44= z-55 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2 ?

1. ( 143,-3, 223 )
2. (2,3, 13 )
3. ( 83,-1, 13 )
4. (- 53,-7,1 )

Let L1: x-13= y-1-1= z+10 and L2: x-22= y0= z+4 , , be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1,1,-1) on L2, then the value of 26 (PB)2 is .

Let a line pass through two distinct points P(-2,-1,3) and Q, and be parallel to the vector 3 i+2 j+2 k. If the distance of the point Q from the point R(1,3,3) is 5, then the square of the area of PQR is equal to :

1. 148
2. 144
3. 136
4. 140

The perpendicular distance, of the line x-12= y+2-1= z+32 from the point P(2,-10,1), is :

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

The shortest distance between the lines x - 34 = y + 7-11 = z - 15 and x - 53 = y - 9-6 = z + 21 is:

1. 185 563
2. 187 563
3. 178 563
4. 179 563

Let the line L intersect the lines x - 2 = -y = z - 1, 2(x + 1) = 2(y - 1) = z + 1 and be parallel to the line x - 23 = y - 11 = z - 22. Then which of the following points lies on L ?

1. (- 13, 1, -1 )
2. (- 13, -1, 1 )
3. (- 13, -1, -1 )
4. (- 13, 1, 1 )

Consider the line L passing through the points (1, 2, 3) and (2, 3, 5). The distance of the point ( 113, 113, 193 ) from the line L along the line 3x - 112 = 3y - 111 = 3z - 192 is equal to

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

The square of the distance of the image of the point (6, 1, 5) in the line x - 13 = y2 = z - 24, from the origin is .

If the shortest distance between the lines L1 : r = (2 + ) i + (1 - 3 ) j + (3 + 4 ) k, R L2 : r = 2(1 + ) i + 3(1 + ) j + (5 + ) k, R is m n, where (m, n) = 1, then the value of m + n equals

1. 384
2. 387
3. 390
4. 377

If the shortest distance between the lines x - 2 = y - 43 = z - 34 and x - 24 = y - 46 = z - 78 is 13 29, then a value of is :

1. 1325
2. 1
3. -1
4. - 1325

Let P( , , ) be the image of the point Q(1, 6, 4) in the line x1 = y - 12 = z - 23. Then 2 + + is equal to

The shortest distance between the lines x - 32 = y + 15-7 = z - 95 and x + 12 = y - 11 = z - 9-3 is

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

Let P be the point (10, -2, -1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, -5, 11) and (-6, 7, -5). Then the length of the line segment PQ is equal to .

Let P( , , ) be the image of the point Q(3, -3, 1) in the line x - 01 = y - 31 = z - 1-1 and R be the point (2, 5, -1). If the area of the triangle PQR is and 2 = 14K, then K is equal to :

1. 18
2. 81
3. 72
4. 36

If the shortest distance between the lines x - 3 = y - 2-1 = z - 11 and x + 2-3 = y + 52 = z - 44 is 44 30, then the largest possible value of | | is equal to .

If the line 2 - x3 = 3y - 24 + 1 = 4 - z makes a right angle with the line x + 33 = 1 - 2y6 = 5 - z7, then 4 + 9 is equal to :

1. 4
2. 13
3. 5
4. 6

Let d be the distance of the point of intersection of the lines x + 63 = y2 = z + 11 and x - 74 = y - 93 = z - 42 from the point (7, 8, 9). Then d2 + 6 is equal to :

1. 75
2. 78
3. 72
4. 69

Let ( , , ) be the image of the point (8, 5, 7) in the line x - 12 = y + 13 = z - 25. Then + + is equal to :

1. 16
2. 20
3. 18
4. 14

Let the point (-1, , ) lie on the line of the shortest distance between the lines x + 2-3 = y - 24 = z - 52 and x + 2-1 = y + 62 = z - 10. Then ( - )2 is equal to .

Let the point, on the line passing through the points P(1,-2,3) and Q(5,-4,7), farther from the origin and at a distance of 9 units from the point P, be ( , , ). Then 2+ 2+ 2 is equal to :

1. 150
2. 155
3. 160
4. 165

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 .