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.

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).