Question

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

Answer 1

The formula for the area ‘A’ encompassed by three points, `(x_1,y_1),(x_2,y_2),and (x_3,y_3)` is given by the formula, 

`triangle=1/2[(x_1y_2+x_2y_3+x_3y_1)-(x_2y_1+x_3y_2+x_1y_3)]` 

If three points are collinear the area encompassed by them is equal to 0.

The three given points are(a,b),(a-a₁ b-b₁)and(a-a₁,b-b₁) If they are collinear then the area enclosed by them should be 0. 

`triangle =1/2[ab_1+a_1(b-b_1)+(a-a_1)b)-(a_1b+(a-a_)b_1+a(b-b_1))]` 

`0=1/2[(ab_1+a_1b-a_1b_1+ab-a_1b)-(a_1b+ab1-a_-1b_1-ab-ab_1)]` 

`0=1/2[ab_1+a_1b-a_1b_1+ab-a_1b-a_1b-a_1b+a_1b_1-ab+ab_1]` 

`0=ab_1-a_1b`

`ab_1=a_1b`   

 Hence we have proved that for the given conditions to be satisfied we need to have 

`a_1b=ab_1 `