UPSC NDA Math Solved Model Paper 8

Given: The lines

x−1

−3

=

y−2

2k

=

z−3

2

‌‌

x−1

−3

=

y−2

2k

=

z−3

2

and

x−1

3k

=

y−5

1

=

z−6

−5

‌‌

x−1

3k

=

y−5

1

=

z−6

−5

are mutually perpendicular.
As we know that, if a, b, c are the direction ration ratios of a line passing through the point (x1,y1,z1), then the equation of line is given by:

x−x1

a

=

y−y1

b

=

z−z1

c

‌‌

x−x1

a

=

y−y1

b

=

z−z1

c

Let L1:

x−1

−3

=

y−2

2k

=

z−3

2

‌

x−1

−3

=

y−2

2k

=

z−3

2

and L2=

x−1

3k

=

y−5

1

=

z−6

−5

‌‌

x−1

3k

=

y−5

1

=

z−6

−5

Now by comparing L1 and L2 with

x−x1

a

=

y−y1

b

=

z−z1

c

‌‌‌

x−x1

a

=

y−y1

b

=

z−z1

c

we get
⇒a1=−3,b1=2k,c1=2, a2=3k,b2=1 and c2=−5
As we know that if two lines are perpendicular then a1.a2+b1.b2+c1.c2=0
⇒a1.a2+b1.b2+c1.c2 =−9k+2k−10=0
⇒ - 7k = 10
⇒ k = - 10/7
Hence, option A is the correct answer