Manipal Entrance Test 2015 Math Paper

The equation of the plane through the intersection of the planes x+y+z=1 and 2x+3y−z+4=0 is
(x+y+z−1)+ λ(2x+3y−z+4)=0
or (2λ+1)x+(3λ+1)y +(1−λ)z+4λ−1=0 .....(i)
It is parallel to X-axis.
i.e.

x

1

=

y

0

=

z

0

∴ 1(2λ+1)+0×(3λ+1) +0(1−λ)=0
⇒ λ=−

1

2

On substituting λ=−

1

2

in Eq. (i), we get
(2x−

1

2

+1)x+(3×

−1

2

+1 )y +(1+

1

2

) z+4×(−

1

2

)−1=0
⇒

−1

2

y+

3

2

z−3=0
⇒ y−3z+6=0
which is the equation of the required plane.