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# If $x,y,z$ are different from zero and $\Delta =\begin{vmatrix}a&b-y&c-z\\a-x&b&c-z\\a-x&b-y&c\end{vmatrix}=0$ then the value of the expression $\large\frac{a}{x}+\frac{b}{y}+\frac{c}{z}$ is

$(a)\;0\qquad(b)\;-1\qquad(c)\;1\qquad(d)\;2$

Given :
$\Delta =\begin{vmatrix}a&b-y&c-z\\a-x&b&c-z\\a-x&b-y&c\end{vmatrix}=0$
Apply $R_1\rightarrow R_1-R_2$
$\qquad R_2\rightarrow R_2-R_3$
$\Rightarrow x\{cy+z(b-y)\}+y\{0+z(a-x)\}=0$
$\Rightarrow ayz+bxz+cxy=2xyz$
$\large\frac{a}{x}+\frac{b}{y}+\frac{c}{z}$$=2$
Hence (d) is the correct answer.