Question

Show that the points (3, 4), (−5, 16), (5, 1) are collinear.

Answer 1

Suppose, the given points are $A\left(3,4\right),B\left(-5,16\right),C\left(5,1\right)$.

We have,
$$\begin{gathered} \overrightarrow{AB}=\mathrm{Position}\mathrm{vector}\mathrm{of}B-\mathrm{Position}\mathrm{vector}\mathrm{of}A \\ =-5\hat{i}+16\hat{j}-3\hat{i}-4\hat{j} \\ =-8\hat{i}+12\hat{j} \\ =-4\left(2\hat{i}-3\hat{j}\right) \\ \overrightarrow{BC}=\mathrm{Position}\mathrm{vector}\mathrm{of}C-\mathrm{Position}\mathrm{vector}\mathrm{of}B \\ =5\hat{i}+\hat{j}+5\hat{i}-16\hat{j} \\ =10\hat{i}-15\hat{j} \\ =5\left(2\hat{i}-3\hat{j}\right) \end{gathered}$$
Therefore, $\overrightarrow{AB}=-\frac{4}{5}\overrightarrow{BC}$
Hence, the given points are collinear.