Q. If $a \neq p,b \neq q,\ c \neq r\$and $\left| \begin{matrix} p & b & c \\ p + a & q + b & 2c \\ a & b & r \end{matrix} \right| = 0,$then $\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}$is equal to
A) 0
B) 1
C) 2
D) 3
Explanation
(c)
We have, $\left| \begin{matrix}
p & b & c \\
p + a & q + b & 2c \\
a & b & r
\end{matrix} \right| = 0$
$\Rightarrow \left| \begin{matrix}
p & b & c \\
p & b & c \\
a & b & r
\end{matrix} \right| + \left| \begin{matrix}
p & b & c \\
a & q & c \\
a & b & r
\end{matrix} \right| = 0$
$\Rightarrow 0 + \left| \begin{matrix}
p & b & c \\
a & q & c \\
a & b & r
\end{matrix} \right| = 0$
$\Rightarrow p(qr - bc) - b(ar - ac) - c(ab - aq) = 0$
$\Rightarrow - pqr + pbc + bar + acq = 0$
On simplifying, we get
$\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c} = 2$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.