Given two lines 2l+m−n=0 ‌⇒‌‌n=2l+m‌ and ‌l2−2m2+n2=0 ‌⇒‌‌l2−2m2+(2l+m)2=0 ‌⇒‌‌l2−2m2+4l2+4lm+m2=0 ‌⇒‌‌5l2+4lm−m2=0 ‌⇒‌‌5(‌
l
m
)2+4(‌
l
m
)−1= Let x=‌
l
m
,5x2+4x−1=0 ‌⇒‌‌x=‌
−4±√42−4(9)(−1)
2(9)
‌⇒‌‌‌
−4±√36
10
=‌
−4±6
10
‌∴‌‌x1=‌
2
10
‌ and ‌x2=‌
−10
10
=−1 Case I‌
l
m
=‌
1
5
⇒‌‌m=5l So, n=2l+m=2l+5l=7l Direction cosine ( l1,m1,n1 ) are proportional to ( l,5l,7l ), i.e., ( 1,5,7 ) Case II ‌
l
m
=−1 ⇒‌‌m=−l So, n=2l+m=2l−l=l Direction cosine ( l2,m2,n2 ) are proportional to ( l,−l,l ) i.e., (1,−1,1) Now, ‌cos‌θ=‌