Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202116 Mar 2021Morning ShiftMathematicsThree Dimensional GeometryActual

If for a > 0 , the feet of perpendiculars from the points A a , - 2 a , 3 and B 0 , 4 , 5 on the plane l x + m y + n z = 0 are points C 0 , - a , - 1 and D respectively, then the length of line segment C D is equal to :

Options

  1. A31
  2. B41
  3. C55
  4. D66

Correct answer

D. 66

Step-by-step solution

C lies on plane ⇒ - m a - n = 0 ⇒ m n = - 1 a … . ( 1 ) Also, CA → will be parallel to the normal to the plane So, CA →   ‖   l i ^ + m j ^ + n k ^ ⇒ a - 0 l = - a m = 4 n ⇒ m n = - a 4 . . . ( 2 ) From ( 1 ) and ( 2 ) - 1 a = - a 4 ⇒ a 2 = 4 ⇒ a = 2 (since a > 0 ) From ( 2 ) , m n = - 1 2 Let m = - t ⇒ n = 2 t 2 l = - 2 - t ⇒ l = t So plane : t x - y + 2 z = 0 B D = 6 6 = 6    C ≡ 0 , - 2 , - 1 C D = B C 2 - B D 2

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