Prove that the points (a, b),(a 1 ,b 1 ) and (a - a 1 , b - b 1 ) are collinear if ab 1 = a 1 b

Question

Philoid · Accepted Answer

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)=_1/2 |x₁(y₁−y3)+x₁(y3−y₁)+x3(y₁−y₁)|

=|a(b₁−(b−b₁))+a₁((b−b₁)−b)+(a−a₁)(b−b₁)|

= |a(b₁−b+b₁)+a₁(b−b₁−b)+a(b−b₁)−a₁(b−b₁)|

=|−ab−a₁b₁+ab−ab₁+a₁b+a₁b₁|

=|−(ab₁−a₁b)|

= (ab₁−a₁b)

This gives, ab₁−a₁b=0

∴ ab₁ = a₁b