UPSC NDA Math Model Paper 1

Let A (a, b, c) is the foot of a perpendicular of point P (2, 2, 2)
Given plane is x - y - z – 1 = 0.
So, PA will be normal to the given plane, so direction ratios of PA will be proportional to (1, -1, -1).
∵ PA passes from (2, 2, 2) and have direction ratios (1, -1, -1).
∴ Equation of line PA=

x−2

1

=

y−2

−1

=

z−2

−1

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