Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

If the image of the point P 1 , - 2,3 in the plane 2 x + 3 y - 4 z + 22 = 0 measured parallel to the line 20 x = 5 y = 4 z is point Q , then the value of | PQ | 2 is

Correct answer

168

Step-by-step solution

Since, distance P Q is measured parallel to the line 20 x = 5 y = 4 z i.e. x 1 = y 4 = z 5 . ⇒ Direction ratios of line P Q are 1 , 4 , 5 Equation of P Q ⇒ x - 1 1 = y + 2 4 = z - 3 5 = λ , (say) ⇒ A point on the line is λ + 1,4 λ - 2,5 λ + 3 ≡ M , (say) Point M lies on the plane 2 x + 3 y - 4 z + 22 = 0 ⇒ 2 λ + 1 + 3 4 λ - 2 - 4 5 λ + 3 + 22 = 0 ⇔ - 6 λ + 6 = 0 ⇒ λ = 1 ⇒ M = 2,2 , 8 M is the midpoint of P Q ⇒ PQ = 2 PM ⇒ PQ = 2 2 - 1 2 + 2 + 2 2 + 8 - 3 2 = 2 1 + 16 + 25 = 2 42 ⇒ | PQ | 2 = 4 × 42 = 168

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 NTA Abhyas JEE Main PYQs