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