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Question
Find the value of \[\lambda\] so that the points (1, −5), (−4, 5) and \[\lambda\] are collinear.
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Solution
If the points (1, −5), (−4, 5) and \[\left( \lambda, 7 \right)\] are collinear, then
\[\begin{vmatrix}1 & - 5 & 1 \\ - 4 & 5 & 1 \\ \lambda & 7 & 1\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}1 & - 5 & 1 \\ - 5 & 10 & 0 \\ \lambda & 7 & 1\end{vmatrix} = 0 \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ \Rightarrow \begin{vmatrix}1 & - 5 & 1 \\ - 5 & 10 & 0 \\ \lambda - 1 & 12 & 0\end{vmatrix} = 0 \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ \Rightarrow ∆ = \begin{vmatrix}- 5 & 10 \\ \lambda - 1 & 12\end{vmatrix} = 0\]
\[ \Rightarrow - 60 - 10\left( \lambda - 1 \right) = 0\]
\[ \Rightarrow - 60 - 10\lambda + 10 = 0\]
\[ \Rightarrow - 10\lambda = 50\]
\[ \Rightarrow \lambda = - 5\]
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