Question

Find the direction cosines of two lines which are connected by the relations $l+m+n=0$ and $mn-2nl-2lm=0.$

Answer 1

Given relation, $l+m+n=0$

$\Rightarrow l=-m-n$....(1)

Given, $mn-2nl-2lm=0$

$mn-2n(-m-n)-2m(-m-n)=0$ [From (1)]

$mn+2mn+2n^2+2m^2+2mn=0$

$2m^2+5mn+2n^2=0$
$2m^2+4mn+mn+2n^2=0$

$2m(m+2n)+n(m+2n)=0$

$(2m+n)(m+2n)=0$

$n=-2m$ ,$n=\frac{-m}{2}$.....(2)

Since, $l=-m-n$

$l=-m+2m$ [From (2)]

$\Rightarrow l=m$

$\Rightarrow \frac{l}{1}=\frac{m}{1}=\frac{n}{-2}$

Since, from (2), $m=-2n$

$l=2n-n$

$\Rightarrow l=n$

$\Rightarrow \frac{l}{1}=\frac{m}{-2}=\frac{n}{1}$

Hence, direction cosines are $<1,1,-2>$ and $<1,-2,1>$