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

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

Solution:

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

$\left|\begin{array}{lll}a & 0 & 1 \\ 0 & b & 1 \\ 1 & 1 & 1\end{array}\right|=0$

$\Rightarrow\left|\begin{array}{ccc}a & 0 & 1 \\ -a & b & 0 \\ 1 & 1 & 1\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$

$\Rightarrow\left|\begin{array}{ccc}a & 0 & 1 \\ -a & b & 0 \\ 1-a & 1 & 0\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$

$\Rightarrow \Delta=\left|\begin{array}{cc}-a & b \\ 1-a & 1\end{array}\right|=0$

$\Rightarrow-a-b(1-a)=0$

$\Rightarrow a+b=a b$