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JEE Main202331 Jan 2023Evening ShiftMathematicsThree Dimensional GeometryActual

The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the point A , B , C is ( 2 , a , 4 ) , a ∈ N . If the volume of the tetrahedron O A B C is 144 unit 3 , then which of the following points is NOT on P ?

Options

  1. A( 0 , 4 , 4 )
  2. B( 3 , 0 , 4 )
  3. C( 0 , 6 , 3 )
  4. D( 2 , 2 , 4 )

Correct answer

B. ( 3 , 0 , 4 )

Step-by-step solution

Plotting the diagram of the given data we get, As ( 2 , a , 4 ) is the foot of perpendicular from origin, So, direction ratio of normal vector will be ( 2 - 0 , a - 0 , 4 - 0 ) ≡ ( 2 , a , 4 ) So, equation of plane P will be, 2 ( x - 2 ) + a ( y - a ) + 4 ( z - 4 ) = 0 ⇒ 2 x + a y + 4 z = 20 + a 2 ⇒ x 20 + a 2 2 + y 20 + a 2 a + z 20 + a 2 4 = 1 Now Volume of tetrahedron O A B C is given by, V = 1 6 · O A · O B · O C ⇒ 1 6 20 + a 2 2 20 + a 2 a 20 + a 2 4 = 144 ⇒ 20 + a

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