First slide

Planes in 3D

Question

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 $P_{1}$ and $P_{2}$ . If the distance of the point (0,1,0) from $P_{3}$ is 1 and the distance of a point $(α, β, γ)$ from $P_{3}$ is 2, then which of the following relations is (are) true?

Moderate

A

$2 α + β + 2 γ + 2 = 0$
B

$2 α - β + 2 γ + 4 = 0$
C

$2 α + β - 2 γ - 10 = 0$
D

$2 α - β + 2 γ - 8 = 0$

Solution

Required plane be $(x + z - 1) + λ y = 0$
$\frac{| λ - 1 |}{\sqrt{λ^{2} + 1 + 1}} = 1 \Rightarrow λ = - \frac{1}{2}$
$P_{3} : 2 x - y + 2 z - 2 = 0$
distance of $P_{3}$ from $(α, β, γ)$ is 2
$\frac{| 2 α - β = 2 γ - 2 |}{\sqrt{4 + 1 + 4}} = 2$
$2 α - β + 2 γ - 8 = 0, 2 α - β + 2 γ + 4 = 0$

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