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Manipal MET2010MathematicsStraight Lines

If t₁, t₂ and t₃ are distinct, the points (t₁, 2 a t₁+a t₁^3 ), (t₂, 2 a t₂+a t₂^3 ) , (t₃, 2 a t₃+a t₃^3 ) are collinear, if

Options

  1. At₁ t₂ t₃=1
  2. Bt₁+t₂+t₃=t₁ t₂ t₃
  3. Ct₁+t₂+t₃=0
  4. Dt₁+t₂+t₃=-1

Correct answer

C. t₁+t₂+t₃=0

Step-by-step solution

The given points are collinear, if | array lll t₁ & 2 a t₁+a t₁^3 & 1 t₂ & 2 a t₂+a t₂^3 & 1 t₃ & 2 a t₃+a t₃^3 & 1 array |=0 a | array lll t₁ & 2 t₁+t₁^3 & 1 t₂ & 2 t₂+t₂^3 & 1 t₃ & 2 t₃+t₃^3 & 1 array |=0 Applying R₂ R₂-R₁, R₃ R₃-R₁ , we get | array ccc t₁ & 2 t₁+t₁^3 & 1 t₂-t₁ & 2 (t₂-t₁ )+ (t₂^3-t₁^3 ) & 0 t₃-t₁ & 2 (t₃-t₁ )+ (t₃^3-t₁^3 ) & 0 array |=0 (t₂-t₁ ) (t₃-t₁ ) | array ccc t₁ & 2 t₁+t₁^3 & 1 1 & 2+t₂^2+t₁^2+t₂ t₁ & 0 1 & 2+t₃^2+t₁^2+t₃ t₁ & 0 array |=0 (t₂-t₁ ) (t₃-t₁ ) (t₃-t₂ ) (t₃+t₂+t₁ )=0 t₁+t₂+t₃=

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