Question

If the points A(m, −1), B(2, 1) and C(4, 5) are collinear, find the value of m.

Answer 1

The given points are A(m, −1), B(2, 1) and C(4, 5).

Now,

$\overrightarrow{\mathrm{AB}}=\left(2\hat{i}+\hat{j}\right)-\left(m\hat{i}-\hat{j}\right)=\left(2-m\right)\hat{i}+2\hat{j}$

$\overrightarrow{\mathrm{AC}}=\left(4\hat{i}+5\hat{j}\right)-\left(m\hat{i}-\hat{j}\right)=\left(4-m\right)\hat{i}+6\hat{j}$

If A, B, C are collinear, then

$$\begin{gathered} \overrightarrow{\mathrm{AB}}=\lambda \overrightarrow{\mathrm{AC}} \\ \Rightarrow \left(2-m\right)\hat{i}+2\hat{j}=\lambda \left[\left(4-m\right)\hat{i}+6\hat{j}\right] \\ \Rightarrow 2-m=\lambda \left(4-m\right)\mathrm{and}2=6\lambda \left(\mathrm{Equating}\mathrm{coefficients}\mathrm{of}\hat{i}\mathrm{and}\hat{j}\right) \\ \Rightarrow \lambda =\frac{1}{3} \end{gathered}$$
and
$$\begin{gathered} 2-m=\frac{1}{3}\left(4-m\right) \\ \Rightarrow 6-3m=4-m \\ \Rightarrow 2m=2 \\ \Rightarrow m=1 \end{gathered}$$

Thus, the value of m is 1.