Question

The vertices of a triangle are [at₁t₂, a(t₁ + t₂)], [at₂t₃, a(t₂ + t₃)] and [at₃t₁, a(t₃ + t₁)]. Find the orthocentre of the triangle.

Answer 1

See Fig.

Let A, B and C be the vertices

A ≡ [at₁t₂, a(t₁ + t₂)]

B ≡ [at₂t₃, a(t₂ + t₃)]

C ≡ [at₃t₁, a(t₃ + t₁)]

The slope of BC is

The slope of altitude AD = −t₃. The equation of AD is

a(t₁ + t₂) = –t₃(x – at₁t₂)  ......(1)

Similarly, the equation of altitude BE is

y – a(t₂ + t₃) = –t₁(x – at₂t₃)    ....(2) 

The coordinates of orthocentre O are obtained by simultaneously solving Eqs. (1) and (2). Subtracting Eq. (2) from Eq. (1), we get 

a(t₃ – t₁) = −x(t₃ – t₁) ⇒ x = –a

Subtracting x = −a in Eq. (1), we get

y = a(t₁ + t₂) – t₃(–a – at₁t₂)

y = a(t₁ + t₂ + t₃ + t₁t₂t₃)

The coordinates of the orthocentre does not depend on the values of t₁, t₂ and t₃.