UPSC NDA Math Model Paper 4

Let A (a, b, c) is foot of perpendicular of point P (-1, -2, 3)
Given plane is x - y + 3z = 2.
So, PA will be normal to the given plane, so direction ratios of PA will be proportional to (1, -1, 3).
∵ PA passes from (-1, -2, 3) and have direction ratios (1, -1, 3).
∴ Equation of line PA=

x+1

1

=

y+2

−1

=

z−3

3

=r( say )
So, point A (a, b, c) in the form of r is (r−1,−r−2,3r+3)
Since A (a, b, c), lies on the given plane, so,Plane x−y+3z=2⇒1.(r−1)–1.(−r−2) +3(r+3)–2=0
∴r=

−8

5

So, foot of perpendicular A(r−1,−r−2,3r+3) will be (

−13

5

,

−2

5

,

−9

5

)