Question

If $\left|\begin{array}{ccc}p& q-y& r-z\\ p-x& q& r-z\\ p-x& q-y& r\end{array}\right|=0$, 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
$\Delta =\left|\begin{array}{ccc}x& -y& 0\\ 0& y& -z\\ p-x& q-y& r\end{array}\right|=0$
$\Rightarrow x\left(yr+qz-yz\right)+y\left(0+pz-zx\right)=0$
$\Rightarrow xyr+xqz-xyz+pyz-xyz=0$
$\Rightarrow pyz+qzx+rxy=2xyz$
$\Rightarrow \frac{p}{x}+\frac{q}{y}+\frac{r}{z}=2$