UPSC NDA Math Model Paper 8

Given: The lines

x

1

=

y−2

λ

=

z+3

3

and

x−2

2

=

y−6

3

=

z−3

4

are coplanar
Here, we have to find the value of λ
As we know that, if two lines

x−x1

a1

=

y−y1

b1

=

z−z1

c1

and

x−x2

a2

=

y−y2

b2

=

z−z2

c2

are coplanar then |

x2−x1	y2−y1	z2−z1
a1	b1	c1
a2	b2	c2

| =0
Here, x1=0,y1=2,z1=−3,a1=1, b1=λ,c1=3
Similarly, x2=2,y2=6,z2=3,a2=2,b2=3 and c2=4
⇒|

2	4	6
1	λ	3
2	3	4

|=0
⇒2×(4λ−9)−4×(4−6)+6 ×(3−2λ)=0
⇒8λ−18+8+18−12λ=0
⇒λ=2
Hence. option B is the correct answer.