Using determinants prove that the points (a, b), (a', b) and (a -… - Vidyadip

Question

Using determinants prove that the points $(a, b), (a', b)$ and $(a - a', b - b')$ are collinear if $ab' = a'b.$

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Answer

$\begin{vmatrix}\text{a}&\text{b}&1\\\text{a}'&\text{b}'&1\\\text{a}-\text{a}'&\text{b}-\text{b}'&1\end{vmatrix}=\begin{vmatrix}\text{a}&\text{b}&1\\\text{a}'-\text{a}&\text{b}'-\text{b}&0\\\text{a}-\text{a}'&\text{b}-\text{b}'&1\end{vmatrix} \ [$Applying $R_2 \rightarrow R_2 - R_1]$
$=\begin{vmatrix}\text{a}&\text{b}&1\\\text{a}'-\text{a}&\text{b}'-\text{b}&0\\-\text{a}'&-\text{b}'&0\end{vmatrix}\ [$Applying $R_3 \rightarrow R_3 - R_1]$
$=\begin{vmatrix}\text{a}'-\text{a}&\text{b}'-\text{b}\\-\text{a}'&-\text{b}'\end{vmatrix}$
$=-\text{b}'(\text{a}'-\text{a})+\text{a}'(\text{b}'-\text{b})$
$=-\text{b}'\text{a}'+\text{b}'\text{a}+\text{a}'\text{b}'-\text{a}'\text{b}$
$=\text{b}'\text{a}-\text{a}'\text{b}$
If the points are collinear then $\triangle=0$
$\text{a}\text{b}'-\text{a}'\text{b}=0$
Thus, $\text{a}\text{b}'=\text{a}'\text{b}$

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