Question

Using determinants show that the following points are collinear:

(3, −2), (8, 8) and (5, 2)

Answer 1

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

\[∆ = \begin{vmatrix}3 & - 2 & 1 \\ 8 & 8 & 1 \\ 5 & 2 & 1\end{vmatrix} = 0\] 
\[ = \begin{vmatrix}3 & - 2 & 1 \\ 5 & 10 & 0 \\ 5 & 2 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\] 
\[ = \begin{vmatrix}3 & - 2 & 1 \\ 5 & 10 & 0 \\ 2 & 4 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\] 
\[ = \begin{vmatrix}5 & 10 \\ 2 & 4\end{vmatrix} = 20 - 20 = 0\]

Thus the points are colinear.