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Three Dimensional Geometry

JEE Main 2025 (Online) 7th April Morning ShiftMCQMedium📈 Topic in 36% papers+4−1
Ques. 1JEE Main 2025, 7 April (Morning Shift)

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)

Answer

(1) (7, 15, 13)

Solution

Let P(1, 1, 1). Line L1 passes through A(1, -1, 1) with direction (2, 3, 4), and line L2 through B(3, 4, 0) with direction (1, 2, 1).

Since L passes through P and meets L1, it lies in the plane containing P and L1. Here AP = (0, 2, 0) and the normal is (2, 3, 4) (0, 2, 0) = (-8, 0, 4) (2, 0, -1), so this plane 1 is 2(x-1) - (z-1) = 0, i.e. 2x - z = 1.

Similarly L lies in the plane containing P and L2: BP = (-2, -3, 1) and the normal is (1, 2, 1) (-2, -3, 1) = (5, -3, 1), so 2 is 5(x-1) - 3(y-1) + (z-1) = 0, i.e. 5x - 3y + z = 3.

Hence the direction of L is (2, 0, -1) (5, -3, 1) = (-3, -7, -6) (3, 7, 6), and

L: x-13 = y-17 = z-16 = , i.e. (1+3 , 1+7 , 1+6 )

= 2 gives (7, 15, 13), which therefore lies on L.

Check: 2(7) - 13 = 1 and 5(7) - 3(15) + 13 = 3 ✓. The other points do not satisfy both plane equations.

Answer: (1)

Keep practising — Three Dimensional Geometry

Ques. 2JEE Main 2025, 7 April (Morning Shift)

If the shortest distance between the lines x-12 = y-23 = z-34 and x1 = y = z-51 is 5 6, then the sum of all possible values of is

  1. 32
  2. 3
  3. -3
  4. - 32
Ques. 3JEE Main 2025, 7 April (Evening Shift)

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
Ques. 4JEE Main 2025, 7 April (Evening Shift)

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
Ques. 5JEE Main 2025, 4 April (Morning Shift)

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
Ques. 6JEE Main 2025, 4 April (Morning Shift)

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
Ques. 7JEE Main 2025, 4 April (Evening Shift)

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
Ques. 8JEE Main 2025, 4 April (Evening Shift)

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
Ques. 9JEE Main 2025, 3 April (Morning Shift)

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
Ques. 10JEE Main 2025, 3 April (Morning Shift)

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
Ques. 11JEE Main 2025, 3 April (Evening Shift)

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

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