In `R^(3)`, consider the planes `P_(1):y=0` and `P_(2),x+z=1.` Let `P_(3)` be a plane, different from `P_(1)` and `P_(2)` which passes through the intersection of `P_(1)` and `P_(2)`, If the distance of the point (0,1,0) from `P_(3)` is 1 and the distance of a point `(alpha,beta,gamma)` from `P_(3)` is 2, then which of the following relation(s) is/are true? (a) `2alpha + beta + 2gamma +2 = 0 ` (b) `2alpha -beta + 2gamma +4=0` (c) `2alpha + beta - 2gamma- 10 =0` (d) `2alpha- beta+ 2gamma-8=0`

Options

A. `2alpha+beta+2gamma+2=0`
B. `2alpha-beta+2gamma+4=0`
C. `2alpha+beta-2gamma-10=0`
D. `2alpha-beta+2gamma-8=8`

Answer

B, D

Explanation

The plane passing through the intersection of planes `P_(1):y=0` and `P_(2):x+z=1` is
`P_(3):(x+z-1)+lamday=0` or `x+lamday+z-1=0` ………..i
It is given that the distances of the point `(0,1,0)` and `(alpha, beta, gamma)` from i are 1 and 2 respectively.
`:.|(0+lamda+0-1)/(sqrt(1+lamda^(2)+1))|=1` and `|(alpha+lamda beta+gamma-1)/(sqrt(1+lamda^(2)+1))|=2`
`implies(|lamda-1|)/(sqrt(lamda^(2)+2))=1` and `(|alpha+lamda beta+gamma-1|)/(sqrt(lamda^(2)+2))=2`
`implies(lamda-1)^(2)=lamda^(2)+2` and `|alpha+lamda beta+gamma-1|=2sqrt(lamda^(2)+2)`
`implieslamda=-1/2` and `|alpha+lamda beta+gamma-1|-2sqrt(lamda^(2)+2)`
`implies|2alpha-beta+2gamma-2|=6`
`implies2alpha-beta+2gamma=+-2alpha-beta+2gamma-8=0`
or `2alpha-beta+2gamma+4=0`