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

Question

Using determinants show that the following points are collinear:
$(3, -2), (8, 8)$ and $(5, 2)$

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Answer

If the points $(3, -2), (8, 8)$ and $(5, 2)$ collinear, then
$\triangle=\begin{vmatrix}3&-2&1\\8&8&1\\5&2&1\end{vmatrix}=0$
$=\begin{vmatrix}3&-2&1\\5&10&0\\5&2&1\end{vmatrix} [$Applying $R_2 \rightarrow R_2 - R_1]$
$=\begin{vmatrix}3&-2&1\\5&10&0\\2&4&0\end{vmatrix} [$Applying $R_3 \rightarrow R_3 - R_1]$
$=\begin{vmatrix}5&10\\2&4\end{vmatrix}$
$=20-20$
$=0$
Thus, points are collinear.

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