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

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

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

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 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

Let the values of for which the shortest distance between the lines x-12 = y-23 = z-34 and x- 3 = y-44 = z-55 is 1 6 be 1 and 2. Then the radius of the circle passing through the points (0, 0), ( 1, 2) and ( 2, 1) is

1. 3
2. 5 23
3. 23
4. 4

If the shortest distance between the lines x-12 = y-23 = z-34 and x1 = y = z-51 is 5 6, then the sum of all possible values of is

1. 32
2. 3
3. -3
4. - 32

Let the shortest distance between the lines x-33 = y- -1 = z-31 and x+3-3 = y+72 = z- 4 be 3 30. Then the positive value of 5 + is

1. 42
2. 40
3. 48
4. 46

Let the values of p, for which the shortest distance between the lines x+13= y4= z5 and r=(p i+2 j+ k)+ (2 i+3 j+4 k) is 1 6, be a, b, (a<b). Then the length of the latus rectum of the ellipse x2a2+ y2b2=1 is :

1. 32
2. 9
3. 18
4. 23

Line L1 passes through the point (1, 2, 3) and is parallel to z-axis. Line L2 passes through the point ( , 5, 6) and is parallel to y-axis. Let for = 1, 2, 2 < 1, the shortest distance between the two lines be 3. Then the square of the distance of the point ( 1, 2, 7) from the line L1 is

1. 25
2. 32
3. 40
4. 37

The line L1 is parallel to the vector a = -3 i + 2 j + 4 k and passes through the point (7, 6, 2) and the line L2 is parallel to the vector b = 2 i + j + 3 k and passes through the point (5, 3, 4). The shortest distance between the lines L1 and L2 is :

1. 23 38
2. 21 38
3. 23 57
4. 21 57

If the square of the shortest distance between the lines x-21= y-12= z+3-3 and x+12= y+34= z+5-5 is mn, where m, n are coprime numbers, then m+n is equal to :

1. 14
2. 6
3. 21
4. 9

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 )

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

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

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

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

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

If the shortest distance between the lines x-41= y+12= z-3 and x- 2= y+14= z-2-5 is 6 5, then the sum of all possible values of is :

1. 10
2. 5
3. 7
4. 8