Question

Using determinants show that the following points are collinear:

(5, 5), (−5, 1) and (10, 7)

Answer 1

If the points  (5, 5), (−5, 1) and (10, 7) are collinear, then

\[∆ = \begin{vmatrix}5 & 5 & 1 \\ - 5 & 1 & 1 \\ 10 & 7 & 1\end{vmatrix} = 0\] 
\[ = \begin{vmatrix}5 & 5 & 1 \\ - 10 & - 4 & 0 \\ 10 & 7 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \begin{vmatrix}5 & 5 & 1 \\ - 10 & - 4 & 0 \\ 5 & 2 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\] 
\[ = \begin{vmatrix}- 10 & - 4 \\ 5 & 2\end{vmatrix} = - 20 + 20 = 0\]

Thus, these points are colinear.