If the points $(a_{1}, b_{1}), (a_{2}, b_{2})$ and $[(a_{1} + a_{2}), (b_{1} + b_{2})]$ collinear, show that $a_{1} b_{2} = a_{2} b_{1}$

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Solution

given points $(a_{1}, b_{1}) (a_{2}, b_{2}) (a_{1} + a_{2}, b_{1} + b_{2})$

for the points to be collinear the slope must be equal

$\frac{b_{2} - b_{1}}{a_{2} - a_{1}} = \frac{b_{1} + b_{2} - b_{2}}{a_{1} + a_{2} - a_{2}}$

$\frac{b_{2} - b_{1}}{a_{2} - a_{1}} = \frac{b_{1}}{a_{1}}$

$b_{2} a_{1} - a_{1} b_{1} = a_{2} b_{1} - a_{1} b_{1}$

$∴ a_{1} b_{2} = a_{2} b_{1}$

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