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Q.

In R3, consider the planes P1:y=0 and .P2:x+z=1  Let P3 be a plane, different from P1 and P2, which passes through the intersection of P1 and P2. If the distance of the point0,1,0  from P3  is 1 and the distance of a point α,β,γ from P3 is 2, then which of the following relations is(are) true?

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a

2α+β+2γ+2=0

b

2α−β+2γ+4=0

c

2α+β−2γ−10=0

d

2α−β+2γ−8=0

answer is B.

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Detailed Solution

Equation of the plane is  P3≡x+z−1+λy=0            Distance from  0,1,0 to the plane is 1 ⇒ 0+0−1+λ11+1+λ2=1   ⇒ λ−12=λ2+2  ⇒λ2+1−2λ=λ2+2 ⇒ −2λ=1  ⇒λ=−12          ∴Equation of the plane is x+z−1-y2=0  ⇒ 2x−y+2z−2=0  Distance from α,β,γ is ⇒2α−β+2γ−21+4+4=2 ⇒2α−β+2γ=2±6=8  or  −4
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