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Question
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
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Solution
If the points (3, −2), (x, 2) and (8, 8) are collinear, then
\[\begin{vmatrix}3 & - 2 & 1 \\ x & 2 & 1 \\ 8 & 8 & 1\end{vmatrix} = 0\]
\[ ∆ = \begin{vmatrix}3 & - 2 & 1 \\ x & 2 & 1 \\ 8 & 8 & 1\end{vmatrix}\]
\[ = \begin{vmatrix}3 & - 2 & 1 \\ x - 3 & 4 & 0 \\ 8 & 8 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \begin{vmatrix}3 & - 2 & 1 \\ x - 3 & 4 & 0 \\ 5 & 10 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \begin{vmatrix}x - 3 & 4 \\ 5 & 10\end{vmatrix}\]
\[ = 10x - 30 - 20\]
\[ ∆ = 10x - 50\]
\[ ∆ = 0 \left[\text{ Given }\right]\]
\[ \Rightarrow 10x - 50 = 0\]
\[ \Rightarrow 10x = 50\]
\[ \Rightarrow x = 5\]
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