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 $\frac{m}{x-m}-1+\frac{y}{y-n}-1+\frac{r}{z-r}$ is

• A. $4$
• B. $2$
• C. $0$
• D. $-2$

Answer 1

The correct option is D $-2$

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

$=-1+1-1-1=-2$