Question

If the lines p₁x + q₁y = 1, p₂x + q₂y = 1 and p₃x + q₃y = 1 be concurrent, show that the points (p₁, q₁), (p₂, q₂) and (p₃, q₃) are collinear.

Answer 1

Given: 

p₁x + q₁y = 1 

p₂x + q₂y = 1 

p₃x + q₃y = 1 

To prove: 

The points (p₁, q₁), (p₂, q₂) and (p₃, q₃) are collinear.

Concept Used: 

If three lines are concurrent then determinant of equation is zero. 

Explanation: 

The given lines can be written as follows: 

p₁ x + q₁ y – 1 = 0 … (1) 

p₂ x + q₂ y – 1 = 0 … (2) 

p₃ x + q₃ y – 1 = 0 … (3) 

It is given that the three lines are concurrent.

Hence proved, This is the condition for the collinearity of the three points, (p₁, q₁), (p₂, q₂) and (p₃, q₃).