Question

If a, b, c are all different and the equations
$ax+a^{2}y+(a^3+1)=0$
$bx+b^{2}y+(b^3+1)=0$
$cx+c^{2}y+(c^3+1)=0$
are consistent then prove that $abc+1=0$

Answer 1

For consistency $D=0$

$\therefore \left|\begin{array}{ccc}a& a^2& a^3+1\\ b& b^2& b^3+1\\ c& c^2& c^3+1\end{array}\right|=0$

Hence we have $1+abc=0.$