Question

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.

Answer 1

Given:

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

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 (at²_2,2at₂),(at²_1,2at₁) and (at²_3,2at₃)