Question

The vertices of a triangle are

[at₁t₂, a(t₁ + t₂)], [at₂t₃, a(t₂ + t₃)], [at₃t₁, a(t₃ + t₁)]

Find the orthocentre of the triangle.

Answer 1

Let ABC be a triangle whose vertices are A[at₁t₂, a(t₁ + t₂)], B[at₂t₃, a(t₂ + t₃)], and C[at1t₃, a(t₁ + t₃)]

Then,

So, the equation of a line through A perpendicular to BC is

and the equation of a line through B perpendicular to AC is

The point of intersection of Eqs. (i) and (ii), is the orthocentre.

On subtracting Eq. (ii) from Eq. (i), we get x = - a.

On putting x = - a in Eq. (i), we get

Hence, the coordinates of the orthocentre are