Question

The condition for the equation $x^2+(\lambda +\mu )xy+{\lambda \mu y}^2+x+\mu y=0$ to represent pair of parallel lines and the distance between them are

• A. $\lambda +\mu =0,\sqrt{1+{\mu }^2}$
• B. $\lambda +\mu =0,\frac{1}{\sqrt{1+{\mu }^2}}$
• C. $\lambda =\mu ,\sqrt{1+{\mu }^2}$
• D. $\lambda =\mu ,\frac{1}{\sqrt{1+{\mu }^2}}$

Answer 1

Answer: (D)

$\lambda =\mu ,\frac{1}{\sqrt{1+{\mu }^2}}$

$x^2+(\lambda +\mu )xy+\lambda \mu y^2+x+\mu y$--(1)

Comparing 1 with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

a=1, h=$\frac{\lambda +\mu }{2}$, b=$\lambda \mu$, g=$\frac{1}{2}$, f=$\frac{\mu }{2}$ ,c=0

Angle between the lines is $\alpha$ then,

$tan\left(\alpha \right)=\frac{2\sqrt{h^2-ab}}{a+b}$

If lines are parallel then $tan\left(\alpha \right)$=0

=>$\sqrt{h^2-ab}=0$

=>$h^2-ab=0$

=>$\frac{(\lambda +\mu {)}^2}{4}=\lambda \mu$

=>${\lambda }^2+2\lambda \mu +{\mu }^2=4\lambda \mu$

=>${\lambda }^2-2\lambda \mu +{\mu }^2=0$=>${\lambda }^2+2\lambda \mu +{\mu }^2=4\lambda \mu$

=>$(\lambda -\mu {)}^2=0$

=>$\lambda =\mu$

The distance between the lines is

$d=\frac{2\cdot \sqrt{g^2-ac}}{a(a+b)}$

$d=\frac{2\cdot \sqrt{\frac{1}{4}-0}}{1+{\mu }^2}$

$d=\frac{1}{1+{\mu }^2}$

Hence,Option D is Correct.