UPSC NDA Math Model Paper 6

Given: The lines

x−3

3

=

y+4

2

=

z−1

λ

and

x+1

3

=

y−2

2

=

z

1

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

−4	6	−1
3	2	λ
3	2	1

|=0
⇒ -4 × (2 - 2λ) - 6 × (3 - 3λ) - 1 × (6 - 6) = 0
⇒ - 8 + 8λ - 18 + 18λ = 0
⇒ λ = 1
Hence. option C is the correct answer.