Three Dimensional Geometry

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

Let the shortest distance between the lines x-33 = y- -1 = z-31 and x+3-3 = y+72 = z- 4 be 3 30. Then the positive value of 5 + is

1. 42
2. 40
3. 48
4. 46

Let A and B be two distinct points on the line L : x - 63 = y - 72 = z - 7-2. Both A and B are at a distance 2 17 from the foot of perpendicular drawn from the point (1, 2, 3) on the line L. If O is the origin, then OA OB is equal to

1. 49
2. 21
3. 47
4. 62

Let A be the point of intersection of the lines L1: x-71= y-50= z-3-1 and L2: x-13= y+34= z+75. Let B and C be the points on the lines L1 and L2 respectively such that AB=AC= 15. Then the square of the area of the triangle ABC is :

1. 63
2. 57
3. 60
4. 54

Let the values of p, for which the shortest distance between the lines x+13= y4= z5 and r=(p i+2 j+ k)+ (2 i+3 j+4 k) is 1 6, be a, b, (a<b). Then the length of the latus rectum of the ellipse x2a2+ y2b2=1 is :

1. 32
2. 9
3. 18
4. 23

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

Line L1 passes through the point (1, 2, 3) and is parallel to z-axis. Line L2 passes through the point ( , 5, 6) and is parallel to y-axis. Let for = 1, 2, 2 < 1, the shortest distance between the two lines be 3. Then the square of the distance of the point ( 1, 2, 7) from the line L1 is

1. 25
2. 32
3. 40
4. 37

Each of the angles and that a given line makes with the positive y- and z-axes, respectively, is half of the angle that this line makes with the positive x-axis. Then the sum of all possible values of the angle is

1. 2
3. 3 4
4. 3 2

The distance of the point (7, 10, 11) from the line x-41 = y-40 = z-23 along the line x-92 = y-133 = z-176 is

1. 16
2. 12
3. 18
4. 14

Let the vertices Q and R of the triangle PQR lie on the line x+35 = y-12 = z+43, QR = 5 and the coordinates of the point P be (0, 2, 3). If the area of the triangle PQR is mn then :

1. 2m - 5 21 n = 0
2. m - 5 21 n = 0
3. 5m - 21 2 n = 0
4. 5m - 2 21 n = 0

If the image of the point P(1, 0, 3) in the line joining the points A(4, 7, 1) and B(3, 5, 3) is Q( , , ), then + + is equal to :

1. 463
2. 18
3. 13
4. 473

The line L1 is parallel to the vector a = -3 i + 2 j + 4 k and passes through the point (7, 6, 2) and the line L2 is parallel to the vector b = 2 i + j + 3 k and passes through the point (5, 3, 4). The shortest distance between the lines L1 and L2 is :

1. 23 38
2. 21 38
3. 23 57
4. 21 57

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 P be the foot of the perpendicular from the point (1,2,2) on the line L: x-11= y+1-1= z-22. Let the line r= (- i+ j-2 k )+ ( i- j+ k ), , intersect the line L at Q. Then 2(PQ)2 is equal to :

1. 25
2. 27
3. 19
4. 29

If the image of the point (4, 4, 3) in the line x - 12 = y - 21 = z - 13 is ( , , ), then + + is equal to

1. 12
2. 9
3. 7
4. 8

The square of the distance of the point ( 157, 327, 7 ) from the line x+13 = y+35 = z+57 in the direction of the vector i + 4 j + 7 k is:

1. 66
2. 54
3. 41
4. 44

Let in a ABC, the length of the side AC be 6, the vertex B be (1,2,3) and the vertices A, C lie on the line x-63= y-72= z-7-2. Then the area (in sq. units) of ABC is:

1. 42
2. 17
3. 56
4. 21

Let the line passing through the point (-1,2,1) and parallel to the line x-12= y+13= z4 intersect the line x+23= y-32= z-41 at the point P. Then the distance of P from the point Q(4,-5,1) is

1. 5 6
2. 5
3. 5 5
4. 10

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line x-37= y-2-1= z+1-2. Then the area of the right angled triangle PQR, where R is the point (3,-2,1), is

1. 30
2. 9 15
3. 3 30
4. 8 15

If the square of the shortest distance between the lines x-21= y-12= z+3-3 and x+12= y+34= z+5-5 is mn, where m, n are coprime numbers, then m+n is equal to :

1. 14
2. 6
3. 21
4. 9

The distance of the line x-22= y-63= z-34 from the point (1,4,0) along the line x1= y-22= z+33 is :

1. 17
2. 13
3. 15
4. 14

Let L1: x-12= y-23= z-34 and L2: x-23= y-44= z-55 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2 ?

1. ( 143,-3, 223 )
2. (2,3, 13 )
3. ( 83,-1, 13 )
4. (- 53,-7,1 )

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

The perpendicular distance, of the line x-12= y+2-1= z+32 from the point P(2,-10,1), is :

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