Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202429 Jan 2024Morning ShiftMathematicsThree Dimensional GeometryActual

A line with direction ratio 2 , 1 , 2 meets the lines x = y + 2 = z and x + 2 = 2 y = 2 z respectively at the point P and Q . if the length of the perpendicular from the point ( 1 , 2 , 12 ) to the line PQ is l , then l 2 is

Correct answer

65

Step-by-step solution

Let P t , t - 2 , t and Q 2 s - 2 , s , s DR ' s of PQ are 2 , 1 , 2 ⇒ 2 s - 2 - t 2 = s - t + 2 1 = s - t 2 ⇒ 2 s - 2 - t 2 = s - t + 2 1 ⇒ 2 s - 2 - t = 2 s - 2 t + 4 ⇒ t = 6 ⇒ s - t + 2 1 = s - t 2 ⇒ s - 6 + 2 1 = s - 6 2 ⇒ 2 s - 8 = s - 6 ⇒ s = 2 ⇒ P 6 , 4 , 6 and Q 2 , 2 , 2 PQ : x - 2 2 = y - 2 1 = z - 2 2 = λ ⇒ x = 2 λ + 2 , y = λ + 2 and z = 2 λ + 2 Let, F ( 2 λ + 2 , λ + 2 , 2 λ + 2 ) A 1 , 2 , 12 AF → · PQ → = 0 ⇒ 2 2 λ + 1 + 1 λ + 2 2 λ - 10 = 0 ⇒ 4 λ + 2 + λ + 4 λ - 20 = 0 ⇒ λ = 2 So F 6 , 4 , 6 and AF

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