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JEE Advanced2024MathematicsThree Dimensional GeometryActual

Let R be such that the lines L ₁: x+11 1 = y +21 2 = z +29 3 and L₂: x+16 3 = y+11 2 = z+4 intersect. Let R ₁ be the point of intersection of L ₁ and L ₂ . Let O =(0,0,0) , and n denote a unit normal vector to the plane containing both the lines L ₁ and L ₂ . Match each entry in List-I to the correct entry in List-II. The correct option is

Options

  1. A( P ) (3) ( Q ) (4) ( R ) (1) ( S ) (2)
  2. B( P ) (5) ( Q ) (4) ( R ) (1) ( S ) (2)
  3. C( P ) (3) ( Q ) (4) ( R ) (1) ( S ) (5)
  4. D( P ) (3) ( Q ) (1) ( R ) (4) ( S ) (5)

Correct answer

C. ( P ) (3) ( Q ) (4) ( R ) (1) ( S ) (5)

Step-by-step solution

aligned & L ₁: x +11 1 = y +21 2 = z +29 3 = a & L ₂: x +16 3 = y +11 2 = z +4 = b aligned aligned & x=a-11=3 b-16 a-3 b=-5...(1) & y=2 a-21=2 b-11 2 a-2 b=10...(2) & z=3 a-29=b r-4 3 a-b =25...(3) aligned from (1) & (2) a =10, ~b =5 Now from (3) 3(10)-5 =25 =1 aligned & R ₁ (-1,-1,1) & OR ₁=- i - j + k & n = | array lrr i & j & k 1 & 2 & 3 3 & 2 & 1 array |=-4 i -(-8) j -4 k & n =-4 i +8 j +4 k =-4( i -2 j + k ) & n = 4( i -2 j + k ) 4 6 = ( i -2 j + k ) 6 & OR ₁ n = (- i - j + k ) ( i -2 j + k 6 )= 2 6 = 4 6 = 2

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