Question

If the pairs of lines $x^2-2nxy-y^2=0$ and $x^2-2mxy-y^2=0$ are such that one of them represents the bisectors of the angles between the other, then

• A. $mn=1$
• B. $m+n=mn$
• C. $mn=-1$
• D. $m-n=mn$

Answer 1

The correct option is C

$mn=-1$

Explanation for the correct answer:

The equation of angle bisectors for a pair of straight lines represented by $a{x}^2+2hxy+b{y}^2=0$ is given as

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

Comparing the given equation $x^2-2mxy-y^2=0$ with $a{x}^2+2hxy+b{y}^2=0$ we get,

$a=1,h=-m,b=-1$

Substituting these values in the equation of angle of bisector we get

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

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

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

Comparing this equation with $x^2-2nxy-y^2=0$ we get

$\Rightarrow$$\frac{2}{m}=-2n$

$\Rightarrow$$mn=-1$

Hence, option (C) is the correct answer.