Question

If $a=\frac{x}{y-z},b=\frac{y}{z-x}andc=\frac{z}{x-y}$

where x, y, z are not all zero, prove that $1+ab+bc+ca=0$.

Answer 1

The given equations can be written as
$x-ay+az=0$
$bx+y-bz=0$
$cx-cy-z=0$

For a non-trivial solution we have

$\therefore \left|\begin{array}{ccc}1& -a& a\\ b& 1& -b\\ c& -c& -1\end{array}\right|=0$

or $1(-1-bc)+a(-b+bc)+a(-bc-c)=0$
or $-1-ab-bc-ca+abc-abc=0$