Three Dimensional Geometry

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

Let the line L pass through the point (-3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point (-2, r, 1) is 143, then the sum of all possible values of r is :

1. 16
2. 12
3. 6
4. 10

Let the line L1 be parallel to the vector -3 i + 2 j + 4 k and pass through the point (2, 6, 7), and the line L2 be parallel to the vector 2 i + j + 3 k and pass through the point (4, 3, 5). If the line L3 is parallel to the vector -3 i + 5 j + 16 k and intersects the lines L1 and L2 at the points C and D, respectively, then | CD |2 is equal to:

1. 290
2. 171
3. 89
4. 312

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

Let the area of the triangle formed by the lines x + 2 = y - 1 = z, x - 35 = y-1 = z - 11 and x-3 = y - 33 = z - 21 be A. Then A2 is equal to .

Let the line L pass through (1, 1, 1) and intersect the lines x-12 = y+13 = z-14 and x-31 = y-42 = z1. Then, which of the following points lies on the line L?

1. (7, 15, 13)
2. (4, 22, 7)
3. (10, -29, -50)
4. (5, 4, 3)

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

If the equation of the line passing through the point (0, - 12, 0 ) and perpendicular to the lines r = ( i + a j + b k ) and r = ( i - j - 6 k ) + (-b i + a j + 5 k ) is x-1-2 = y+4d = z-c-4, then a + b + c + d is equal to :

1. 13
2. 14
3. 12
4. 10

Consider the lines L1: x - 1 = y - 2 = z and L2: x - 2 = y = z - 1. Let the feet of the perpendiculars from the point P(5, 1, -3) on the lines L1 and L2 be Q and R respectively. If the area of the triangle PQR is A, then 4A2 is equal to :

1. 151
2. 147
3. 139
4. 143