UPSC NDA Math Model Paper 2

Given: The lines

x−1

2

=

y−2

3

=

z−3

4

and

x−2

3

=

y−3

4

=

z−4

λ

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

1	1	1
2	3	4
3	4	λ

|=0
⇒1×(3λ−16)−1×(2λ−12)+1 ×(8−9)=0
⇒3λ−16−2λ+12−1=0
⇒λ=5
Hence. option D is the correct answer.