Three Dimensional Geometry

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

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