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 Here, `P_(3):(x+z-1)+lambday=0` i.e.`" "P_(3):x+lambday+z-1=0` whose distance from (0,1,0) is 1. `:." "(\|0+lambda+0-1\|)/(sqrt(1+lambda^(2)+1))=1` `implies" "\|lambda-1\|=sqrt(lambda^(2)+2)` `implies" "lambda^(2)-2lambda+1=lambda^(2)+2implieslambda=-(1)/(2)` `:."Equation of "P_(3)" is "2x-y+2z-2=0.` `because` Distance from `(alpha,beta,gamma)` is 2. `:." "(\|2alpha-beta+2gamma-2\|)/(sqrt(4+1+4))=2` `implies" "2alpha-beta+2gamma-2=+-6` `implies2alpha-beta+2gamma=8" and "2alpha-beta+2gamma=-4`

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