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Distance Formula

Prove that th...

Question

Prove that the points $(a, b), (a_{1}, b_{1}$ and $(a - a_{1}, b - b_{1})$ are collinear if $a b_{1} = a_{1} b$ .

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Solution

$A (a, b), B (a_{1}, b_{1})$ and $C (a - a_{1} b -, b_{1})$
$=> \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b a_{1} & b_{1} a - a_{1} & b - b_{1} a & b \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = \frac{1}{2} | a b_{1} - b a_{1} + a_{1} b - a_{1} b_{1} - b_{1} a + b_{1} a_{1} + b a - a_{1} b - a b + a b_{1} | = \frac{1}{2} | a_{1} b - a b_{1} | = 0 (A s g i v e n a b_{1} = a_{1} b)$

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