Question

Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear, if `(1/a + 1/b) = 1`.

Answer 1

Consider the points A (a,0), B( 0,b) and C (1,1) .

` Here (x_1=a,y_1=0).(x_2 = 0,y_2=b) and (x_3=1,y_3=1).`

It is given that the points are collinear. So,

`x_1 (y_2-y_3)+x_2(y_3-y_1) +x_3(y_1-y_2) =0`

`⇒  a(b-1)+0(1-0)+1(0-b)=0`

`⇒ ab-a-b=0`

Dividing the equation by ab:

`⇒ 1-1/b-1/a=0`

`⇒ 1-(1/a+1/b)=0`

`⇒(1/a+1/b)=1`

Therefore, the given points are collinear if  `(1/a+1/b)=1`