Question

If $x>m,y>n,z>r(x,y,z>0)$ such that $\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& n& z\end{array}\right|=0$

The value of $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$ is ?

• A. 1
• B. -1
• C. 2
• D. -2

Answer 1

Answer: (C)

2

$\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& n& z\end{array}\right|=0$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$\left|\begin{array}{ccc}x-m& n-y& 0\\ 0& y-n& r-z\\ m& n& z\end{array}\right|=0$
$\Rightarrow (x-m)(y-n)z-n(x-m)(r-z)+m(n-y)(r-z)=0$
$\Rightarrow \frac{z}{z-r}+\frac{n}{y-n}+\frac{m}{x-m}=0$
$\Rightarrow \frac{z}{z-r}=-\frac{m}{x-m}-\frac{n}{y-n}$ ....(i)
Consider, $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$
$=\frac{x}{x-m}+\frac{y}{y-n}-\frac{m}{x-m}-\frac{n}{y-n}$ (using (i))
$=1+1=2$