Three Dimensional Geometry

A line with direction ratios 1, -1, 2 intersects the lines x2= y3= z+13 and x+1-1= y-21= z4 at the points P and Q, respectively. If the length of the line segment PQ is , then 225 2 is equal to:

1. 1024
2. 1014
3. 1104
4. 1204

If the distances of the point (1, 2, a) from the line x - 11 = y2 = z - 11 along the lines L1 : x - 13 = y - 24 = z - ab and L2 : x - 11 = y - 24 = z - ac are equal, then a + b + c is equal to

1. 4
2. 6
3. 7
4. 5

Let the lines L1: r = i + 2 j + 3 k + (2 i + 3 j + 4 k), R and L2: r = (4 i + j) + (5 i + 2 j + k), R, intersect at the point R. Let P and Q be the points lying on lines L1 and L2, respectively, such that | PR| = 29 and | PQ| = 473. If the point P lies in the first octant, then 27(QR)2 is equal to

1. 360
2. 348
3. 320
4. 340

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 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 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 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

Let a line passing through the point (4, 1, 0) intersect the line L1 : x-12 = y-23 = z-34 at the point A( , , ) and the line L2 : x - 6 = y = -z + 4 at the point B(a, b, c). Then vmatrix 1 & 0 & 1 \ & & \ a & b & c vmatrix is equal to

1. 16
2. 6
3. 8
4. 12

Let L1 : x-11 = y-2-1 = z-12 and L2 : x+1-1 = y-22 = z1 be two lines. Let L3 be a line passing through the point ( , , ) and be perpendicular to both L1 and L2. If L3 intersects L1, then |5 - 11 - 8 | equals :

1. 25
2. 20
3. 16
4. 18

Let a = i + 2 j + k and b = 2 i + 7 j + 3 k. Let L1 : r = (- i + 2 j + k) + a, R and L2 : r = ( j + k) + b, R be two lines. If the line L3 passes through the point of intersection of L1 and L2 and is parallel to a + b, then L3 passes through the point :

1. (-1, -1, 1)
2. (2, 8, 5)
3. (8, 26, 12)
4. (5, 17, 4)