UPSC NDA Math Model Paper 5

We know that
The lines

x−1

k

=

y−4

2

=

z−5

1

and

x−2

1

=

y−3

1

=

z−4

−k

are coplanar then
⇒|

x2−x1	y2−y1	z2−z1
a1	b1	c1
a2	b2	c2

| =0
Given The lines

x−1

k

=

y−4

2

=

z−5

1

and

x−2

1

=

y−3

1

=

z−4

−k

are coplanar
Here, x1=1,x2=2,y1=4,y2=3,z1=5,z2=4, a1=k,b1=2,c1=1 a2=1,b2=1,c2=−k
⇒|

1	−1	−1
k	2	1
1	1	−k

|=0
⇒1(−2k−1)+1(−k2−1)−1(k−2)=0
⇒−2k−1−k2−1−k+2=0
⇒−k2−3k=0
⇒k(k+3)=0
Neglect k=0
⇒k=−3
Hence, The lines

x−1

k

=

y−4

2

=

z−5

1

and

x−2

1

=

y−3

1

=

z−4

−k

are coplanar if the value of k = - 3