#### Question

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.

#### Solution

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-a _{1} b-b_{1})*and(a-a

_{1},b-b

_{1}) 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 `