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Question
If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.
Solution
If the points (x, −2), (5, 2), (8, 8) are collinear, then
\[\begin{vmatrix}x & - 2 & 1 \\ 5 & 2 & 1 \\ 8 & 8 & 1\end{vmatrix} = 0\]
\[ ∆ = \begin{vmatrix}x & - 2 & 1 \\ 5 & 2 & 1 \\ 8 & 8 & 1\end{vmatrix}\]
\[ ∆ = \begin{vmatrix}x & - 2 & 1 \\ 5 - x & 4 & 0 \\ 8 & 8 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \begin{vmatrix}x & - 2 & 1 \\ 5 - x & 4 & 0 \\ 8 - x & 10 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \begin{vmatrix}5 - x & 4 \\ 8 - x & 10\end{vmatrix}\]
\[ = 50 - 10x - 32 + 4x\]
\[ = 18 - 6x\]
\[ ∆ = 18 - 6x\]
\[ ∆ = 0 \left[\text{ Given }\right]\]
\[ \Rightarrow 18 - 6x = 0\]
\[ \Rightarrow x = 3\]
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