Question

Using determinants show that the following points are collinear:

(2, 3), (−1, −2) and (5, 8)

Answer 1

If the points (2, 3), (−1, −2) and (5, 8) are collinear, then

\[∆ = \begin{vmatrix}2 & 3 & 1 \\ - 1 & - 2 & 1 \\ 5 & 8 & 1\end{vmatrix} = 0\] 
\[ = \begin{vmatrix}2 & 3 & 1 \\ - 3 & - 5 & 0 \\ 5 & 8 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \begin{vmatrix}2 & 3 & 1 \\ - 3 & - 5 & 0 \\ 3 & 5 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \begin{vmatrix}- 3 & - 5 \\ 3 & 5\end{vmatrix} = - 15 + 15 = 0\]

Thus the points are colinear.