Parabola

Consider a triangle ABC formed by three points t₁ , t₂ , t₃ on the parabola y²=4ax. The focus S is the orthocentre of the triangle.

Calculate : (t₁-1)(t₂-1)(t₃-1)

2 Answers

262

Aditya Bhutra ·2012-03-20 00:24:45

line joining S and t₁ is perp. to line joining t₂ and t₃

thus 2t₁t₁²-1. 2t₂+t₃ = -1

or 4t₁t₁²-1 -t₁ = -(t₁ +t₂ +t₃ ) = k (say)

similarly for t₂ and t₃ .

4tt²-1 -t = k

whose roots are t₁ ,t₂,t₃ .

on simplifying ,

f(t) = t³ +kt² -5t -k =0

thus (t₁-1)(t₂-1)(t₃-1) = f(1) = 1 +k -5 -k = -4

Thanks! I didn't do after second line.

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