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Most Important Selected Qs for JEE AdvancedMathematicsThree Dimensional Geometry

A plane meets the co-ordinates axes in A, B, C such that the centroid of the triangle A B C is the point (1, r, r^2 ) . If all such planes must pass through the point (t, t, t) for all r R- 0 , then t can satisfy

Options

  1. A0 t 4
  2. B1 t 4
  3. C2 t 3
  4. D0 t 4, t 3

Correct answer

C. 2 t 3

Step-by-step solution

Let the equation of the required plane be x a + y b + z c =1 The points A, B, C are (a, 0,0),(0, b, 0) and (0,0, c) and the centroid of A B C is ( a 3 , b 3 , c 3 ) Thus a=3, b=3 r, c=3 r^2, r 0 So, the equation of plane is x 3 + y 3 r + z 3 r^2 =1 It passes through ( t , t , t ) so, ( t -3) r ^2+ rt + t =0 , clearly t 0 for real r, t^2-4(t-3) t 0, t 3 0 t 4, t 3 when t =3, r =-1 , so 0 t 4

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