UPSC NDA Math Model Paper 4

Given: The lines

x−2

1

=

y−4

4

=

z−6

7

and

x+1

3

=

y+3

5

=

z+5

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=2,y1=4,z1=6,a1=1, b1=4,c1=7
Similarly, x2=−1,y2=−3,z2=−5, a2=3,b2=5 and c2=λ
⇒|

−3	−7	−11
1	4	7
3	5	λ

| =0
⇒−3×(4λ−35)+7×(λ−21)−11 ×(5−12)=0
⇒−12λ+105+7λ−147+77=0
⇒λ=7
Hence. option A is the correct answer.