The variable plane `(2lambda+1)x+(3-lambda)y+z=4` always passes through the line

Options

A. `x/0=y/0=(z-4)/1`
B. `x/1=y/2=(z-4)/-3`
C. `x/1=y/1=(z-4)/-7`
D. `x/1=y/2=(z-4)/-7`

Answer

D

Explanation

The plane is `x+3y+z-4+lambda(2x-y)=0`
This is always passes through the line of intersection of the planes `x+3y+z=4=0` and `2x-y=0`
Now, `2x-y=0 rArr x/1=y/2`
`therefore x+3y+z-4=0`
`rArr x+3.2x+z-4=0`
`rArr x+3.2x+z-4=0`
`rArr 7x+z-4=0`
`rArr 7x = -(z-4)`
`rArr x/1=(z-4)/-7`
`therefore` line is `x/1=y/2=(z-4)/-7`