Question

If $\left|\begin{array}{ccc}a& b-y& c-z\\ a-x& b& c-z\\ a-x& b-y& c\end{array}\right|=0,$ then $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=......$

Answer 1

Apply $R_1-R_2$ and $R_2-R_3$.
$\Delta =\left|\begin{array}{ccc}x& -y& 0\\ 0& y& -z\\ a-x& b-y& c\end{array}\right|=0$
or $x\{cy+z(b-y)\}+y\left\{z(a-x)\right\}=0$
or $xyc+zxb+yza=2xyz$
$\therefore \frac{c}{z}+\frac{b}{y}+\frac{a}{x}=2$.