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Question
If \[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0, then using properties of determinants, find the value of \[\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\] , where \[x, y, z \neq\] 0
Solution
\[\begin{vmatrix}x & - y & 0 \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} = 0\]
\[ R_2 \to R_2 - R_3 \]
\[ \Rightarrow \begin{vmatrix}x & - y & 0 \\ 0 & y & - z \\ a - x & b - y & c\end{vmatrix} = 0\]
\[\text{ Expanding along first row, we get }\]
\[x(yc + zb - zy) + y(0 - za + zx) = 0\]
\[ \Rightarrow xyc + xzb - xyz + zya - xyz = 0 \]
\[\text{ Dividing by xyz, we get }\]
\[\frac{c}{z} + \frac{b}{y} - 2 + \frac{a}{x} = 0\]
\[ \therefore \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2\]
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