Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202624 January 2026Morning ShiftMathematicsThree Dimensional GeometryActual

Let the lines L ₁: r = i +2 j +3 k + (2 i +3 j +4 k ), R and L ₂: r =(4 i + j )+ (5 i +2 j + k ), R , intersect at the point R. Let P and Q be the points lying on lines L ₁ and L ₂ , respectively, such that | PR |= 29 and | PQ |= 47 3 . If the point P lies in the first octant, then 27( QR )² is equal to

Options

  1. A348
  2. B340
  3. C320
  4. D360

Correct answer

D. 360

Step-by-step solution

Find R by solving L₁ = L₂ : From 1+2 = 4+5 , 2+3 = 1+2 , 3+4 = we get = -1 , = -1 , so R = (-1, -1, -1) Point P on L₁ : P = (1+2 _P, 2+3 _P, 3+4 _P) with | PR |^2 = 29 4(1+ _P)^2 + 9(1+ _P)^2 + 16(1+ _P)^2 = 29 → 29(1+ _P)^2 = 29 → _P = 0 (first octant) Thus P = (1, 2, 3) Point Q on L₂ : Q = (4+5 _Q, 1+2 _Q, _Q) with | PQ |^2 = 47 3 (3+5 _Q)^2 + (-1+2 _Q)^2 + ( _Q-3)^2 = 47 3 30 _Q^2 + 20 _Q + 19 = 47 3 → 9 _Q^2 + 6 _Q + 1 = 0 → _Q = - 1 3 So Q = ( 7 3 , 1 3 , - 1 3 ) |QR|^2 = 100+16+4 9 = 120 9 = 40 3 27(QR)^2 = 3

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let P be the plane such that it contains the straight line x-1 2 = y-3 3 = z+2 1 and is perpendicular to the plane x + 2y + 3z = 4 . Let P₁ be the plane which passes through the po 2026Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the foot of the perpendicular drawn from the point R(4, -1, 5) to the line L . Another line pass 2026Let the foot of perpendicular from the point ( , 2, 3) on the line x-4 1 = y-9 2 = z-5 1 be the point (1, , 2) . Then the distance between the lines x-1 2 = y-2 3 = z+4 6 and x- 2 2026Let 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 2026The shortest distance between the lines x-4 1 = y-3 2 = z-2 -3 and x+2 2 = y-6 4 = z-5 -5 is : 2026Let the image of the point P(0, -5, 0) in the line x-1 2 = y 1 = z+1 -2 be the point R and the image of the point Q (0, -1 2 , 0 ) in the line x-1 -1 = y+9 4 = z+1 1 be the point S 2026Let the image of the point P(1, 6, a) in the line L: x 1 = y-1 2 = z-a+1 b , b > 0 , be ( a 3 , 0, a+c ) . If S( , , ) , > 0 , is the point on L such that the distance of S from th 2026Let a line L be perpendicular to both the lines L₁: x+1 3 = y+3 5 = z+5 7 and L₂: x-2 1 = y-4 4 = z-6 7 . If is the acute angle between the lines L and L₃: x - 8 7 2 = y - 4 7 1 = 2026 Full Three Dimensional Geometry list All JEE Main PYQs