Question

If the normal at the point ‘t₁‘ on the parabola y² = 4ax meets the parabola again at the point ‘t₂‘ , then prove that t₂ = (t₁ + (2/t₁))

Answer 1

Equation of normal to y² = 4 at’ t’ is y + xt = 2at + at³. 

So equation of normal at ‘t₁’ is y + xt₁ = 2at₁ + at₁³ 

The normal meets the parabola y² = 4ax at ‘t₂’ 

(ie.,) at (at₂², 2at₂) 

⇒ 2at₂ + at₁t₂² = 2at₁ + at₁³ 

So 2a(t₂ – t₁) = at₁³ – at₁t₂² 

⇒ 2a(t₂ – t₁) = at₁(t₁² – t₂²) 

⇒ 2(t₂ – t₁) = t₁(t₁ + t₂)(t₁ – t₂) 

⇒ 2 = -t₁(t₁ + t₂) 

⇒ t₁ + t₂ = -2/t₁

⇒ t₂ = -t₁ - (2/t₁) = -(t₁ + (2/t₁))