Question

If $\begin{vmatrix} p & q-y & r-z \\[0.3em] p-x & q & r-z \\[0.3em] p-x & q-y & r \end{vmatrix}=0 $, then the value of $\frac{p}{x}+\frac{q}{y}+\frac{r}{z}$ is

• A. 0
• B. 1
• C. 2
• D. 4pqr

Answer 1

Answer: (C)

Operate $R_1 - R_2 ; R_2 - R_1$, we get
$\Delta = \begin{vmatrix}x&-y&0\\ 0&y&-z\\ p-x&q-y&r\end{vmatrix}=0$
$ \Rightarrow \,x\left(yr+qz-yz\right)+y\left(0+pz-zx\right)=0$
$ \Rightarrow \, xyr+xqz-xyz +pyz-xyz =0$
$ \Rightarrow \, pyz+qzx+rxy= 2xyz$
$ \Rightarrow \,\frac{p}{x}+\frac{q}{y}+\frac{r}{z}=2$