Using determinants show that the following points are collinear: … - Vidyadip

Question

Using determinants show that the following points are collinear:
$(1, -1), (2, 1)$ and $(4, 5)$

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Answer

If the points $(1, -1), (2, 1)$ and $(4, 5)$ are collinear, then
$\triangle=\begin{vmatrix}1&-1&1\\2&1&1\\4&5&1\end{vmatrix}=0$
$=\begin{vmatrix}1&-1&1\\1&2&0\\4&5&1\end{vmatrix} [$Applying $R_2 \rightarrow R_2 - R_1]$
$=\begin{vmatrix}1&-1&1\\1&2&0\\3&6&0\end{vmatrix} [$Applying $R_3 \rightarrow R_3 - R_1]$
$=\begin{vmatrix}1&2\\3&6\end{vmatrix}$
$=6-6=0$
Thus, these points are collinear.

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