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In R^(3), consider the planes P(1):y=0 a...

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`

Text Solution

Verified by Experts

The correct Answer is:
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.
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