Question

Prove that the points (a, b),(a₁,b₁) and (a - a₁, b - b₁) are collinear if ab₁ = a₁b

Answer 1

Consider the following points A(a,b), B(a₁,b₁), C(a−a₁,b−b₁) 

Since the given points are collinear, we have area(△ABC)=0 

First find the area of area(△ABC) as follows: 

area(△ABC)

= \(\frac{1}2\) |x₁(y₁−y₃)+x₁(y₃−y₁)+x₃(y₁−y₁)| 

= \(\frac{1}2\) |a(b₁−(b−b₁))+a₁((b−b₁)−b)+(a−a₁)(b−b₁)| 

= \(\frac{1}2\) |a(b₁−b+b₁)+a₁(b−b₁−b)+a(b−b₁)−a₁(b−b₁)| 

= \(\frac{1}2\) |−ab−a₁b₁+ab−ab₁+a₁b+a₁b₁| 

= \(\frac{1}2\) |−(ab₁−a₁b)| = (ab₁−a₁b) 

This gives, ab₁−a₁b=0 

∴ ab₁ = a₁b