If the points $(a_{1}, b_{1})$ , $(a_{2}, b_{2})$ and $(a_{1} + a_{2}, b_{1} + b_{2})$ are collinear, then show that $a_{1} b_{2} = a_{2} b_{1}$ .

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Solution

$(a_{1}, b_{1}) (a_{2}, b_{2}) (a_{1} + a_{2}, b_{1} + b_{2})$ points are collinear
Slopes should be equal
$\frac{b_{2} - b_{1}}{a_{2} - a_{1}} = \frac{b_{1} + b_{2} - b_{2}}{a_{1} + a_{2} - a_{2}}$
$\frac{b_{2} - b_{1}}{a_{2} - a_{1}} = \frac{b_{1}}{a_{1}}$
$a_{1} b_{2} - a_{2} b_{1} = a_{2} b_{1} - a_{1} b_{1}$
$a_{1} b_{2} = a_{2} b_{1}$

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