Question

The angle between the straight lines, whose direction cosines are given by the equations $2l+2m-n=0$ and $mn+nl+lm=0,$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\pi -{\cos }^{-1}\left(\frac{4}{9}\right)$
• D. ${\cos }^{-1}\left(\frac{8}{9}\right)$

Answer 1

The correct option is B $\frac{\pi }{2}$

$\because 2l+2m-n=0\cdots \left(i\right)$
and $mn+nl+lm=0\cdots \left(ii\right)$
From equation (i) and (ii)
$(m+l)(2l+2m)+lm=0$
$\Rightarrow 2l^2+5lm+2m^2=0$
$\Rightarrow 2l^2+4lm+lm+2m^2=0$
$\Rightarrow 2l(l+2m)+m(l+2m)=0$
$\therefore (2l+m)(l+2m)=0$
Direction ratios of lines are $(1,-2,-2)$ and $(2,-1,2).$
Here $l_1{l}_2+m_1{m}_2+n_1{n}_2=0$
$\therefore$ Lines are perpendicular to each other.