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Three Dimensional Geometry · Question 43

JEE Main 2025 (Online) 3rd April Morning ShiftMCQMedium📈 Topic in 35% papers+4−1
1

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

Answer

(3) 8

Solution

A on L1: A(1 + 2s, 2 + 3s, 3 + 4s); B on L2: B(6 + t, t, 4 - t)

With P(4, 1, 0): PA = (2s - 3, 3s + 1, 4s + 3), PB = (t + 2, t - 1, 4 - t)

P, A, B collinear ⇒ PA = k PB:

2s - 3 = k(t + 2) … (i), 3s + 1 = k(t - 1) … (ii), 4s + 3 = k(4 - t) … (iii)

(i) − (ii): -s - 4 = 3k; (ii) + (iii): 7s + 4 = 3k

⇒ 7s + 4 = -s - 4 ⇒ s = -1, k = -1

From (i): -5 = -(t + 2) ⇒ t = 3

⇒ A(-1, -1, -1), B(9, 3, 1)

∴ vmatrix 1 & 0 & 1 \ -1 & -1 & -1 \ 9 & 3 & 1 vmatrix = 1(-1 + 3) - 0 + 1(-3 + 9)

= 2 + 6 = 8

Answer: (3)

JEE Main 2025 (Online) 3rd April Morning Shift

Keep practising — Three Dimensional Geometry

2

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
JEE Main 2025 (Online) 3rd April Morning Shift
3

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
JEE Main 2025 (Online) 3rd April Evening Shift
4

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
JEE Main 2025 (Online) 3rd April Evening Shift
5

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
JEE Main 2025 (Online) 2nd April Morning Shift
6

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
JEE Main 2025 (Online) 2nd April Evening Shift
7

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
JEE Main 2025 (Online) 2nd April Evening Shift
8

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
JEE Main 2025 (Online) 29th January Morning Shift
9

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)
JEE Main 2025 (Online) 29th January Morning Shift
10

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
JEE Main 2025 (Online) 28th January Morning Shift
11

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
JEE Main 2025 (Online) 28th January Evening Shift

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