KVPY Sample Paper KVPY Stream-SX Model Paper-3

The plane through the intersection of the planes \[x+y+z=1\] and \[2x+3y-z+4=0\] and parallel to y - axis also passes through the point:

A) \[(-\,3,0,-1)\]

B) \[(3,3,-1)\]

C) \[(3,2,1)\]

D) \[(-\,3,1,1)\]

Correct Answer: C

Solution :

Plane through the intersection of the given planes is of the form

\[(x+y+z\,-\,1)\,+\,\lambda \,(2x+3y-z+4)\,=\,0\]

\[(2\lambda +1)x\,+\,(3\lambda +1)y+\,(1-\lambda )z+\,(4\lambda -1)\,=\,0,\lambda \,\in \,R\]

This is parallel to y-axis \[\Rightarrow \]\[3\lambda \,+\,1\,=\,0\]

\[\therefore \] \[\lambda \,=\,-\,\text{1/3}\]

\[\therefore \]The equation.of the plane is \[x\,+\,4z-\,7\,=\,0\]

It is passing through (3, 2, 1).