Question

lf the $D.R$s of two lines are given by the equations $l+m+n=0$ and $l^2+m^2-n^2=0$, then the angle between the two lines is:

• A. ${60}^o$
• B. ${30}^o$
• C. ${45}^o$
• D. ${90}^o$

Answer 1

The correct option is A ${60}^o$

Given that $l+m=-n$
Substituting $n$ in $l^2+m^2-n^2=0$, we get

$l^2+m^2-(l+m{)}^2=0$
$\Rightarrow lm=0$
$\Rightarrow l=0$ or $m=0$
Substituting in $l+m=-n$, we get

$m=-n$ or $l=-n$
Now, $(l,m,n)=(0,-n,n)$ or $(-n,0,n)$
Therefore, d.r.s are $(0,-1,1)$ or $(-1,0,1)$
Now, angle between two lines $\cos \theta =\frac{(0,-1,1).(-1,0.1)}{\sqrt{{(-1)}^2+1^2}\sqrt{{(-1)}^2+1^2}}=\frac{1}{2}$

$\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{2}\right)$
$\Rightarrow \theta =\frac{\pi }{3}$
Ans: A