Concept: If a,b,c are the direction ration ratios of a line passing through the point (x1,y1,z1), then the equation of line is given by: ‌
x−x1
a
=‌
y−y1
b
=‌
z−z1
c
If a,b,c are the direction ration ratiosof a line then the direction cosine of the line is given by: 1=‌
a
√a2+b2+c2
m=‌
b
√a2+b2+c2
,n=‌
c
√a2+b2+c2
Calculation Given: 1,m,n are the direction cosinesof the line x−1=2(y+3)=1−z The given equation of lines can be re-written as ‌
x−1
1
=‌
y+3
1
2
=‌
z−1
−1
So, by comparing the equation ‌
x−1
1
=‌
y+3
1
2
=‌
z−1
−1
‌‌ with ‌
x−x1
a
=‌
y−y1
b
=‌
z−z1
c
we get ⇒a=1,b=1∕2‌ and ‌c=−1 As we know that, if a,b,c are the direction ration ratios of a line then the direction cosine of the line is given by: 1=‌
a
√a2+b2+c2
m=‌
b
√a2+b2+c2
,n=‌
c
√a2+b2+c2
The direction cosine of the given line is: ⇒1=‌
2
3
,m=‌
1
3
,n=‌
−2
3
⇒14+m4+n4=16∕81+1∕81+16∕81=33∕81=11∕27 Hence, the correct option is 2 .