If the two pair of lines x2 -2mxy - y2 = 0 and x2-2nxy- y2 = 0 are such that one of them represents the bisector of the angles between the other, then

Question

If the two pair of lines x2 -2mxy - y2 = 0 and x2-2nxy- y2 = 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

Tardigrade Admin · Accepted Answer

Bisector of the angles between the lines x2-2 m x y-y2=0 is (x2-y2/x y) =(1-(-1)/-m) ⇒ m x2+2 x y-m y2 =0 But it represents by x2-2 n x y-y2=0 ∴ (m/1)=(2/-2 n) ⇒ m n=-1

Q. If the two pair of lines
$x^{2} - 2 m x y - y^{2} = 0$ and $x^{2} - 2 n x y - y^{2} = 0$
are such that one of them represents the bisector of the angles between the other, then

$mn = 1$

$m + n = mn$

$mn = - 1$

$m - n = mn$

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 $x^{2} - 2 n x y - y^{2} = 0$
$∴ \frac{m}{1} = \frac{2}{- 2 n}$
$\Rightarrow mn = - 1$

Q. If the two pair of lines x2−2mxy−y2=0 and x2−2nxy−y2=0 are such that one of them represents the bisector of the angles between the other, then

mn=1

m+n=mn

mn=−1

m−n=mn