If the bisectors of the pair of lines x 2 - 2 m x y - y 2 = 0 is represented by x 2 - 2 n x y - y 2 = 0 , then

Given equation is $x^2-2mxy-y^2=0 .....\left(1\right)$

Compare the given equation with $a{x}^2+2hxy+b{y}^2=0 ......\left(2\right)$

We have, $a=1,$ $h=-m$ and $b=-1$

Now the equation of the bisector of the angle between the pair of lines in equation $\left(1\right)$ is

$\frac{x^2-y^2}{a-b}=\frac{xy}{h}$

$\Rightarrow \frac{x^2-y^2}{1-\left(-1\right)}=\frac{xy}{{\displaystyle -m}}$

$\Rightarrow -m{x}^2+m{y}^2=2xy$

$\Rightarrow m{x}^2-m{y}^2+2xy=0$

Now on comparing this equation with $x^2-2nxy-y^2=0$

We get, $m=1,n=-1$

So option $\left(1\right)$ is correct.