Mathematics - Determinant Question with Solution | TestHub

$\left|\begin{array}{ccc}\mathrm{p}& \mathrm{q}-\mathrm{y}& \mathrm{r}-\mathrm{z}\\ \mathrm{p}-\mathrm{x}& \mathrm{q}& \mathrm{r}-\mathrm{z}\\ \mathrm{p}-\mathrm{x}& \mathrm{q}-\mathrm{y}& \mathrm{r}\end{array}\right|=0$ Apply ${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_3$ and ${\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_3,$ we get $\left|\begin{array}{ccc}\mathrm{x}& 0& -\mathrm{z}\\ 0& \mathrm{y}& -\mathrm{z}\\ \mathrm{p}-\mathrm{x}& \mathrm{q}-\mathrm{y}& \mathrm{r}\end{array}\right|=0$ $\Rightarrow \mathrm{x}[\mathrm{yr}+\mathrm{z}(\mathrm{q}-\mathrm{y})]-\mathrm{z}[0-\mathrm{y}(\mathrm{p}-\mathrm{x})]=0$ [Expansion along first row $]$ $\Rightarrow xyr+zxq+yzp=2xyz\Rightarrow \frac{p}{x}+\frac{q}{y}+\frac{r}{z}=2$