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Question
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
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Solution
If the points (3, −2), (8, 8) and (5, 2) are collinear, then
\[∆ = \begin{vmatrix}3 & - 2 & 1 \\ 8 & 8 & 1 \\ 5 & 2 & 1\end{vmatrix} = 0\]
\[ = \begin{vmatrix}3 & - 2 & 1 \\ 5 & 10 & 0 \\ 5 & 2 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \begin{vmatrix}3 & - 2 & 1 \\ 5 & 10 & 0 \\ 2 & 4 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \begin{vmatrix}5 & 10 \\ 2 & 4\end{vmatrix} = 20 - 20 = 0\]
Thus the points are colinear.
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