Three Dimensional Geometry

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 .