Question

If the lines $ax+y+1=0$, $x+by+1=0$ and $x+y+c=0$ ($a,b,c$ are distinct and different from each other) are concurrent, then the value of $a+b+c-abc$ is equal to

Answer 1

Given lines are concurrent, so
$\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=0$
$\Rightarrow a(bc-1)-1(c-1)+1(1-b)=0$
$\Rightarrow abc-a-c+1+1-b=0$
$\Rightarrow a+b+c-abc=2$