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Question
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
Solution
If the points (5, 5), (−5, 1) and (10, 7) are collinear, then
\[∆ = \begin{vmatrix}5 & 5 & 1 \\ - 5 & 1 & 1 \\ 10 & 7 & 1\end{vmatrix} = 0\]
\[ = \begin{vmatrix}5 & 5 & 1 \\ - 10 & - 4 & 0 \\ 10 & 7 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \begin{vmatrix}5 & 5 & 1 \\ - 10 & - 4 & 0 \\ 5 & 2 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \begin{vmatrix}- 10 & - 4 \\ 5 & 2\end{vmatrix} = - 20 + 20 = 0\]
Thus, these points are colinear.
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