Question

Prove that the point of intersection of the tangents at ‘t₁‘ and t₂’ on the parabola y² = 4ax is [at₁ t₂, a (t₁ + t₂)]

Answer 1

Equation of the tangent of parabola y² = 4ax be

At t₁ yt₁ = x + at₁²   ........(1)

At t₂ yt₂ yt = x + at₂²   ........(2)

(1) – (2) ⇒ y(t₁ – t₂) = a(t₁² – t₂²)

y(t₁ – t₂) = a(t₁ + t₂)(t₁ – t₂)

y = a(t₁ + t₂)

(1) ⇒ t₁a(t₁ + t₂) = x + at₁²

x = at₁² + at₁t₂ – at₁²

x = at₁t₂

Point of intersection be [at₁t₂, a(t₁ + t₂)]