Question

Prove that the family of lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.

Answer 1

Given: 

Lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary 

To prove: 

The family of lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point. 

Explanation: 

The given family of lines can be written as 

x + 2y + 5 + λ (x − y) = 0 

This line is of the form L₁ + λL₂ = 0, which passes through the intersection of L₁ = 0 and L₂ = 0. 

⇒ x + 2y + 5 = 0 

⇒ x − y = 0 

Now, solving the lines: \(\Big(-\frac{5}{3},-\frac{5}{3}\Big)\)

This is a fixed point. 

Hence proved.