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In R(3) , consider the planes P(1):...

In `R_(3) , ` consider the planes `P_(1):y=0and P_(2) : x+z=1. Let P_(3)` be a plane , different from `P_(1) and P_(2)` , which passes through the interesection of `P_(1) and P_(2) `I fhte distance of the (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 relations 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`

Text Solution

Verified by Experts

The correct Answer is:
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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