Manipal Entrance Test 2014 Math Paper

The equation of any plane through the intersection of the planes,
3x−y+2z−4=0 and x+y+z−2=0, is
3x−y+2z−4)+λ(x+y+z−2)=0 ....(i)
The plane passes through the point (2,2,1) Therefore, this point will satisfy Eq. (i).
∴(3×2−2+2×1−4) +λ(2+2+1−2)=0
⇒ (6−4)+3λ=0
⇒ 2+3λ=0
⇒ λ=

−2

3

On substituting this value of λ in Eq. (i), we get the required plane as
(3x−y+2z−4)−

2

3

(x+y+z−2)=0
⇒ 9x−3y+6z−12−2x −2y−2z+4=0
⇒ 7x−5y+4z−8=0
This is the required equation of the plane.