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Question
If the points A(m, −1), B(2, 1) and C(4, 5) are collinear, find the value of m.
Solution
The given points are A(m, −1), B(2, 1) and C(4, 5).
Now,
\[\overrightarrow{AB} = \left( 2 \hat{i} + \hat{j} \right) - \left( m \hat{i} - \hat{j} \right) = \left( 2 - m \right) \hat{i} + 2 \hat{j}\]
\[\overrightarrow{AC} = \left( 4\hat{i} + 5 \hat{j} \right) - \left( m \hat{i} - \hat{j} \right) = \left( 4 - m \right) \hat{i} + 6 \hat{j}\]
If A, B, C are collinear, then
\[\overrightarrow{AB} = \lambda \overrightarrow{AC} \]
\[ \Rightarrow \left( 2 - m \right) \hat{i} + 2 \hat{j} = \lambda\left[ \left( 4 - m \right) \hat{i} + 6 \hat{j} \right]\]
\[ \Rightarrow 2 - m = \lambda\left( 4 - m \right)\text{ and }2 = 6\lambda \left(\text{ Equating coefficients of }\hat{i}\text{ and }\hat{j} \right)\]
\[ \Rightarrow \lambda = \frac{1}{3}\]
and \[2 - m = \frac{1}{3}\left( 4 - m \right)\]
\[ \Rightarrow 6 - 3m = 4 - m\]
\[ \Rightarrow 2m = 2\]
\[ \Rightarrow m = 1\]
Thus, the value of m is 1.
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