Question

Using determinants show that the following points are collinear:

(1, −1), (2, 1) and (4, 5)

Answer 1

If the points (1, −1), (2, 1) and (4, 5) are collinear, then

\[∆ = \begin{vmatrix}1 & - 1 & 1 \\ 2 & 1 & 1 \\ 4 & 5 & 1\end{vmatrix} = 0\] 
\[ = \begin{vmatrix}1 & - 1 & 1 \\ 1 & 2 & 0 \\ 4 & 5 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\] 
\[ = \begin{vmatrix}1 & - 1 & 1 \\ 1 & 2 & 0 \\ 3 & 6 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \begin{vmatrix}1 & 2 \\ 3 & 6\end{vmatrix} = 6 - 6 = 0\]

Thus, these points are collinear.