Question

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

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

Answer 1

Answer: (C)

Bisector of the angles between the lines
$x^{2}-2 m x y-y^{2}=0$ is
$\frac{x^{2}-y^{2}}{x y} =\frac{1-(-1)}{-m} $
$\Rightarrow m x^{2}+2 x y-m y^{2} =0$
But it represents by $x^{2}-2 n x y-y^{2}=0$
$\therefore \frac{m}{1}=\frac{2}{-2 n} $
$\Rightarrow m n=-1$