NDA Math Three Dimensional Geometry

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 .