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

\[\begin{vmatrix}a & 0 & 1 \\ 0 & b & 1 \\ 1 & 1 & 1\end{vmatrix} = 0\] 
\[ \Rightarrow \begin{vmatrix}a & 0 & 1 \\ - a & b & 0 \\ 1 & 1 & 1\end{vmatrix} = 0 \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\] 
\[ \Rightarrow \begin{vmatrix}a & 0 & 1 \\ - a & b & 0 \\ 1 - a & 1 & 0\end{vmatrix} = 0 \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\] 
\[ \Rightarrow ∆ = \begin{vmatrix}- a & b \\ 1 - a & 1\end{vmatrix} = 0\] 
\[ \Rightarrow - a - b\left( 1 - a \right) = 0\] 
\[ \Rightarrow a + b = ab\]