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

If the distance of the point P(43, , ), < 0, from the line r = 4 i - k + (2 i + 3 k), R along a line with direction ratios 3, -1, 0 is 13 10, then 2 + 2 is equal to

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 a line L passing through the point P(1, 1, 1) be perpendicular to the lines x-44 = y-11 = z-11 and x-171 = y-711 = z0. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point S(1, 0, -1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to .

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 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 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)

Let a straight line L pass through the point P(2, -1, 3) and be perpendicular to the lines x-12 = y+11 = z-3-2 and x-31 = y-23 = z+24. If the line L intersects the yz-plane at the point Q, then the distance between the points P and Q is:

1. 10
2. 2
3. 2 3
4. 3

Let a line pass through two distinct points P(-2,-1,3) and Q, and be parallel to the vector 3 i+2 j+2 k. If the distance of the point Q from the point R(1,3,3) is 5, then the square of the area of PQR is equal to :

1. 148
2. 144
3. 136
4. 140

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

Let d be the distance of the point of intersection of the lines x + 63 = y2 = z + 11 and x - 74 = y - 93 = z - 42 from the point (7, 8, 9). Then d2 + 6 is equal to :

1. 75
2. 78
3. 72
4. 69

Let the point, on the line passing through the points P(1,-2,3) and Q(5,-4,7), farther from the origin and at a distance of 9 units from the point P, be ( , , ). Then 2+ 2+ 2 is equal to :

1. 150
2. 155
3. 160
4. 165

Let P and Q be the points on the line x+38= y-42= z+12 which are at a distance of 6 units from the point R(1,2,3). If the centroid of the triangle PQR is ( , , ), then 2+ 2+ 2 is :

1. 18
2. 24
3. 26
4. 36

A line passes through A(4,-6,-2) and B(16,-2,4). The point P(a, b, c), where a, b, c are non-negative integers, on the line AB lies at a distance of 21 units, from the point A. The distance between the points P(a, b, c) and Q(4,-12,3) is equal to .

Let L1: r=( i- j+2 k)+ ( i- j+2 k), R, L2: r=( j- k)+ (3 i+ j+p k), R, and L3: r= ( i+m j+n k), R be three lines such that L1 is perpendicular to L2 and L3 is perpendicular to both L1 and L2. Then, the point which lies on L3 is

1. (1,7,-4)
2. (1,-7,4)
3. (-1,7,4)
4. (-1,-7,4)

Let a line passing through the point (-1,2,3) intersect the lines L1: x-13= y-22= z+1-2 at M( , , ) and L2: x+2-3= y-2-2= z-14 at N(a, b, c). Then, the value of ( + + )2(a+b+c)2 equals .

Let PQR be a triangle with R(-1,4,2). Suppose M(2,1,2) is the mid point of PQ. The distance of the centroid of from the point of intersection of the lines x-20= y2= z+3-1 and x-11= y+3-3= z+11 is

1. 69
2. 99
3. 69
4. 9