First slide

Planes in 3D

Question

$Let α, β, γ, δ be real numbers such that α^{2} + β^{2} + γ^{2} \neq 0 and α + γ = 1 . Suppose the point (3, 2, - 1) is the$
$mirror image of the point (1, 0, - 1) with respect to the plane α x + β y + γ z = δ . Then which of the$
$following statements is/are TRUE?$

Difficult

A

$α + β = 2$
B

$δ - γ = 3$
C

$δ + β = 4$
D

$α + β + γ = δ$

Solution

$R is mid point of PQ$
$\begin{array}{l} ∴ R (2, 1, - 1) and it lies on plane \\ equation of plane is α x + β y + γ z = δ \end{array}$
$\begin{array}{l} ∴ 2 α + β - γ = δ \dots (1) \\ Normal vector to plane is \\ \vec{n} = 2 i + 2 j \end{array}$
$\begin{array}{l} \vec{n} = 2 i + 2 j \\ ∴ \frac{α}{2} = \frac{β}{2} = \frac{γ}{0} = k \\ ∴ α = 2 k, β = 2 k, γ = 0 \dots . . . (2) \\ and α + γ = 1 (given).....(3) \end{array}$
$\begin{array}{l} from (2) and (3) \\ ∴ α = 1, β = 1, γ = 0 \end{array}$
$\begin{array}{l} and from (1) \\ 2 (1) + 1 - 0 = δ \\ δ = 3 \end{array}$
$now:$
$\begin{array}{l} α + β = 2 \\ δ - γ = 3 \\ δ + β = 4 \\ so, A, B, C are correct. \end{array}$

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