Three Dimensional Geometry

Let the image of the point P(1, 6, a) in the line L : x1= y-12= z-a+1b, b>0, be ( a3, 0, a+c ). If S( , , ), >0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2 14, then + + is equal to :

1. 19
2. 20
3. 21
4. 22

Let the image of the point P(0, -5, 0) in the line x-12= y1= z+1-2 be the point R and the image of the point Q (0, -12, 0 ) in the line x-1-1= y+94= z+11 be the point S. Then the square of the area of the parallelogram PQRS is .

The square of the distance of the point P(5, 6, 7) from the line x-22= y-53= z-24 is equal to:

1. 3
2. 5
3. 6
4. 8

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

1. 4
2. 5
3. 6
4. 8

If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line x1= y-11= z-22 is 4, then the sum of all possible values of a is equal to :

1. 11
2. 9
3. 6
4. 4

The square of the distance of the point (-2, -8, 6) from the line x-11= y-12= z-1 along the line x+51= y+5-1= z2 is equal to:

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

If (2 +1, 2-3 , -12 ) is the image of ( , 2 , 1) in the line x-23= y-12= z1, then the possible value(s) of is (are)

1. Only 3
2. Only 3 and -1
3. Only 3, 14 and -1
4. Only 3 and 14

Let a line L passing through the point (1, 1, 1) be perpendicular to both the vectors 2 i+2 j+ k and i+2 j+2 k. If P(a, b, c) is the foot of perpendicular from the origin on the line L, then the value of 34(a+b+c) is :

1. 50
2. 80
3. 100
4. 120

Let the point A be the foot of perpendicular drawn from the point P(a, b, 0) on the line x-12= y-21= z- 3. If the midpoint of the line segment PA is (0, 34, - 14 ), then the value of a2+b2+ 2 is equal to :

1. 1
2. 2
3. 6
4. 9

Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line x - 11 = y2 = z - 11. Then the distance of Q from the line x - 93 = y - 92 = z - 5-2 is

1. 8
2. 7
3. 6
4. 5

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 .

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 ( , , ) 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

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 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

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

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 P be the image of the point Q(7,-2,5) in the line L: x-12= y+13= z4 and R(5,p,q) be a point on L. Then the square of the area of PQR is .

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

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-13= y-1-1= z+10 and L2: x-22= y0= z+4 , , be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1,1,-1) on L2, then the value of 26 (PB)2 is .

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

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 .