UPSC NDA Math Model Paper 9

Given:
A = (1, 0, 3)
B = (4, 7, 1)
C = (3, 5, 3)
Direction ratio of line BC is (-1, -2, 2)
Equation of BC,

x−4

3−4

=

y−7

5−7

=

z−1

3−1

⇒

x−4

−1

=

y−7

−2

=

z−1

2

=k
⇒x=−k+4,y=−2k+7 and z=2k+1
General point on BC be D = (-k + 4, -2k + 7, 2k + 1) and let D be foot of perpendicular.
Direction ratio of line AD will be (-k + 4 - 1), (-2k + 7 - 0), (2k + 1 - 3)
⇒ (-k + 3, -2k + 7, 2k - 2)
Now, since the line BC and AD are perpendicular, l1l2+m1m2+nln2=0
So, (−k+3)×−1+(−2k+7)×−2 +(2k−2)×2=0
⇒k−3+4k−14+4k−4=0
⇒9k=21
⇒k=

7

3

So, D=(

5

3

,

7

3

,

17

3

)