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Three Dimensional Geometry

JEE Main 2024 (Online) 27th January Evening ShiftNumericalMedium📈 Topic in 71% papers+4
Ques. 1JEE Main 2024, 27 January (Evening Shift)

The lines x-22= y-2= z-716 and x+34= y+23= z+21 intersect at the point P. If the distance of P from the line x+12= y-13= z-11 is l, then 14l2 is equal to .

Answer

108

Solution

Line 1: (2+2 , -2 , 7+16 )

Line 2: (-3+4 , -2+3 , -2+ )

x: 2+2 =-3+4 ; y: -2 =-2+3 =1, =- 12

(z: 7-8=-1=-2+1 ✓) P =(1,1,-1)

[figure]

AP=2 i-2 k, AP2=8

Projection AM= AP d| d|= 4+0-2 14= 2 14

l2=AP2-AM2=8- 414= 10814

14l2=108

Answer: 108

Keep practising — Three Dimensional Geometry

Ques. 2JEE Main 2026, 8 April (Evening Shift)

Let the foot of perpendicular from the point ( , 2, 3) on the line x-41= y-92= z-51 be the point (1, , 2). Then the distance between the lines x-12= y-23= z+46 and x- 2= y- 3= z+56 is equal to :

  1. 127
  2. 1457
  3. 1467
  4. 1437
Ques. 3JEE Main 2026, 8 April (Evening Shift)

Let a line L1 pass through the origin and be perpendicular to the lines L2 : r=(3+t) i+(2t-1) j+(2t+4) k and L3 : r=(3+2s) i+(3+2s) j+(2+s) k, t, s R. If (a, b, c), a Z, is the point on L3 at a distance of 17 from the point of intersection of L1 and L2, then (a+b+c)2 is equal to .

Ques. 4JEE Main 2026, 6 April (Morning Shift)

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
Ques. 5JEE Main 2026, 6 April (Morning Shift)

Let a line L be perpendicular to both the lines L1 : x+13= y+35= z+57 and L2 : x-21= y-44= z-67. If is the acute angle between the lines L and L3 : x- 872= y- 471= z2, then is equal to :

  1. 32 2
  2. 52 2
  3. 53 2
  4. 43 2
Ques. 6JEE Main 2026, 6 April (Evening Shift)

The shortest distance between the lines x-41= y-32= z-2-3 and x+22= y-64= z-5-5 is :

  1. 5 66
  2. 2 5
  3. 3 5
  4. 4 5
Ques. 7JEE Main 2026, 6 April (Evening Shift)

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 .

Ques. 8JEE Main 2026, 5 April (Morning Shift)

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
Ques. 9JEE Main 2026, 5 April (Morning Shift)

The square of the distance of the point of intersection of the lines r= ( i+ j- k )+ (a i- j ), a 0 and r= (4 i- k )+ (2 i+a k ) from the origin is :

  1. 5
  2. 10
  3. 17
  4. 26
Ques. 10JEE Main 2026, 5 April (Evening Shift)

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
Ques. 11JEE Main 2026, 5 April (Evening Shift)

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

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