Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsThree Dimensional Geometry

A plane P₁ contains the line L₁ : x-2 1 = y-1 2 = z+1 -1 and is perpendicular to the plane P₂ : x + y - z = 5 . A line L passes through the point (1, 1, 1) and is perpendicular to the plane P₁ . The vertices B and C of a triangle ABC lie on the line L , such that BC = 4 . If the coordinates of vertex A are (3, 4, 5) , then the area (in sq. units) of ABC is :

Options

  1. A4 11
  2. B7 2
  3. C2 11
  4. D2 29

Correct answer

C. 2 11

Step-by-step solution

The plane P₁ contains the line L₁ with direction ratios (1, 2, -1) and is perpendicular to P₂ with normal vector (1, 1, -1) . The normal vector n ₁ to the plane P₁ is given by the cross product of these two vectors: n ₁ = vmatrix i & j & k 1 & 2 & -1 1 & 1 & -1 vmatrix = i (-2 + 1) - j (-1 + 1) + k (1 - 2) = - i + 0 j - k Thus, the direction ratios of the normal to P₁ are (1, 0, 1) . The line L is perpendicular to P₁ , so its direction ratios are also (1, 0, 1) . Since L passes through (1, 1, 1) , its equation is:

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