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 R₂ → R₂ - R₁]
$=\begin{vmatrix}1&-1&1\\1&2&0\\3&6&0\end{vmatrix}$ [Applying R₃ → R₃ - R₁]
$=\begin{vmatrix}1&2\\3&6\end{vmatrix}$
$=6-6=0$
Thus, these points are collinear.

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