Question

I. If the d.cs of two non-parallel lines satisfy $1+m+n=0$ and $l^2+m^2-n^2=0$, then the angle between the line is $\frac{\pi }{3}$
II. If the d.rs of two non-parallel lines are $(0,\lambda ,-\lambda )$ and $(\mu ,0,-\mu )$, then angle between the lines is $\frac{\pi }{3}(\lambda >0,\mu >0)$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is B Both A and R are true and R is not the correct explanation of A

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 $\cos \theta =\frac{(0,-1,1).(-1,0.1)}{\sqrt{{(-1)}^2+1^2}\sqrt{{(-1)}^2+1^2}}=\frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{3}$
Both A and R are correct but R is not the correct explanation for A.
Ans: B