Question

Using determinants prove that the points (a, b), (a', b') and (a − a', b − b') are collinear if ab' = a'b.

Answer 1

$$\begin{gathered} \left|\begin{array}{ccc}a& b& 1\\ a\text{'}& b\text{'}& 1\\ a-a\text{'}& b-b\text{'}& 1\end{array}\right| \\ \Rightarrow ∆=\left|\begin{array}{ccc}a& b& 1\\ a\text{'}-a& b\text{'}-b& 0\\ a-a\text{'}& b-b\text{'}& 1\end{array}\right|\left[\mathrm{Applying}{R}_2\to R_2-R_1\right] \\ \Rightarrow ∆=\left|\begin{array}{ccc}a& b& 1\\ a\text{'}-a& b\text{'}-b& 0\\ -a\text{'}& -b\text{'}& 0\end{array}\right|\left[\mathrm{Applying}{R}_3\to R_3-R_1\right] \\ \Rightarrow ∆=\left|\begin{array}{cc}a\text{'}-a& b\text{'}-b\\ -a\text{'}& -b\text{'}\end{array}\right| \\ \Rightarrow ∆=-b\text{'}\left(a\text{'}-a\right)+a\text{'}\left(b\text{'}-b\right) \\ =-b\text{'}a\text{'}+b\text{'}a+a\text{'}b\text{'}-a\text{'}b \\ =b\text{'}a-a\text{'}b \end{gathered}$$

If the points are collinear, then ∆ = 0. So,
ab' − a'b = 0

Thus, ab' = a'b