UPSC NDA Math Model Paper 1

Given: The lines

x−3

1

=

y+2

−4

=

z

5

and

x−4

1

=

y−3

−4

=

z−a

5

are coplanar
Here, we have to find the value of a
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=−2,z1=0,a1=1, b1=−4,c1=5
Similarly, x2=4,y2=3,z2=a,a2=1,b2=−4 and c2=5
⇒|

1	5	a
1	−4	5
1	−4	5

|=0
⇒1×(−20+20)−5×(5−5)+a ×(−4+4)=0
⇒ a can be any real number
Hence. option D is the correct answer.