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

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Solution

The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆ = \frac{1}{2} |(x_{1} y_{2} + x_{2} y_{3} + x_{3} y_{1}) - (x_{2} y_{1} + x_{3} y_{2} + x_{1} y_{3})|$

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

The three given points are, and. If they are collinear then the area enclosed by them should be 0.
$∆ = \frac{1}{2} |(a b_{1} + a_{1} (b - b_{1}) + (a - a_{1}) b) - (a_{1} b + (a - a_{1}) b_{1} + a (b - b_{1}))| 0 = \frac{1}{2} |(a b_{1} + a_{1} b - a_{1} b_{1} + a b - a_{1} b) - (a_{1} b + a b_{1} - a_{1} b_{1} + a b - a b_{1})| 0 = \frac{1}{2} |a b_{1} + a_{1} b - a_{1} b_{1} + a b - a_{1} b - a_{1} b - a b_{1} + a_{1} b_{1} - a b + a b_{1}| 0 = a b_{1} - a_{1} b a b_{1} = a_{1} b$

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

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