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

Answer: (A)

$mn+1=0$

Answer 2

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.
$\therefore$ $\frac{1}{1}=\frac{\frac{2}{m}}{-n}$
$\Rightarrow$ $mn=-1\Rightarrow mn+1=0$