Find the coordinates of the vertices of a triangle, the equations of whose sides are :

y(t₁ + t₂) = 2x + 2at₁t₂. y(t₂ + t₃) = 2x + 2at₂t₃ and, y(t₃ + t₁) = 2x + 2at₁t₃.

Given:

y (t₁ + t₂) = 2x + 2a t₁t₂, y (t₂ + t₃) = 2x + 2a t₂t₃ and y (t₃ + t₁) = 2x + 2a t₁t₃

To find:

Point of intersection of pair of lines.

Concept Used:

Point of intersection of two lines.

Explanation:

2x − y (t₁ + t₂) + 2a t₁t₂ = 0 … (1)

2x − y (t₂ + t₃) + 2a t₂t₃ = 0 … (2)

2x − y (t₃ + t₁) + 2a t₁t₃ = 0 … (3)

Solving (1) and (2) using cross - multiplication method:

Solving (1) and (3) using cross - multiplication method:

Solving (2) and (3) using cross - multiplication method:

Hence, the coordinates of the vertices of the triangle are