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

Answer 1

\[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0

\[R_1 \to R_1 - R_2 \] 
\[\begin{vmatrix}x & - y & 0 \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} = 0\] 
\[ R_2 \to R_2 - R_3 \] 
\[ \Rightarrow \begin{vmatrix}x & - y & 0 \\ 0 & y & - z \\ a - x & b - y & c\end{vmatrix} = 0\] 
\[\text{ Expanding along first row, we get }\] 
\[x(yc + zb - zy) + y(0 - za + zx) = 0\] 
\[ \Rightarrow xyc + xzb - xyz + zya - xyz = 0 \] 

\[xyc + xzb - 2xyz + zya = 0\] 
\[\text{ Dividing by xyz, we get }\] 
\[\frac{c}{z} + \frac{b}{y} - 2 + \frac{a}{x} = 0\] 
\[ \therefore \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2\]