Question

If $\left|\begin{array}{ccc}p& b& c\\ a& q& c\\ a& b& r\end{array}\right|=0$ find the value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ and $(p\ne a,q\ne b,r\ne c)$.

Answer 1

$\left|\begin{array}{ccc}p& b& c\\ a& q& c\\ a& b& r\end{array}\right|=0$ is possible when all three rows are equal.

Consider $p=a=a;b=q=b;c=c=r$

$\frac{a}{a-a}+\frac{q}{q-q}+\frac{r}{r-r}=0$