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`

In `R^(3)`, consider the planes `P_(1):y=0` and `P_(2),x+z=1.` Let `P_

1.	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`
Answer» Correct Answer - b, d Clearly, plane `P_(3)` is `P_(2)+ lamdaP_(1)=0`. `rArr" "x+lamday+z-1=0` Distance of this plane from point `(0, 1, 0)` is 1. `rArr " "(0+lamda+0-1)/(sqrt(1+lamda^(2)+1))= pm 1` `therefore " "lamda= - (1)/(2)` Thus, equation of `P_(3)` is `2x-y+2z-2=0`. Distance of this plane from point `(alpha, beta, gamma)` is 2. `rArr" "\|(2alpha-beta+2gamma-2)/(3)\|=2` `rArr" "2alpha-beta+2gamma= 2pm 6` Thus, option (b), (d) are correct.