Question

If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.

Answer 1

If the points (a, 0), (0, b) and (1, 1) are collinear, then

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