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=0 $
• B. $ mn-1=0 $
• C. $ \frac{1}{m}+\frac{1}{n}=0 $
• D. $ \frac{1}{m}-\frac{1}{n}=0 $

Answer 1

The Correct Option is A

Equation of the bisectors of the angle between the lines
$ {{x}^{2}}-2nxy-{{y}^{2}}=0 $ are given by $ \frac{{{x}^{2}}-{{y}^{2}}}{1-(-1)}=\frac{xy}{-m}\Rightarrow {{x}^{2}}+\frac{2}{m}xy-{{y}^{2}}=0 $ ..(i) Since, (i) and $ {{x}^{2}}-2nxy-{{y}^{2}}=0 $
represents the same pair of lines.
$ \therefore $ $ \frac{1}{1}=\frac{\frac{2}{m}}{-n} $
$ \Rightarrow $ $ mn=-1\Rightarrow mn+1=0 $