Let α, β, γ, δ be real numbers such that α2+β2+γ2 ≠ 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?

Question

Let α, β, &gamma;, δ be real numbers such that α2+β2+&gamma;2 ≠ 0 and α+&gamma;=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+&gamma; z=δ. Then which of the following statements is/are TRUE? (A) α+β=2 (B) δ-&gamma;=3 (C) δ+β=4 (D) α+β+&gamma;=δ

Tardigrade Admin · Accepted Answer

Direction ratio of line joining (3,2,-1) &(1,0,-1) is (2,2,0) proportional to (α, β, &gamma;) ⇒ &gamma;=0, α+&gamma;=1, α=1, β=1 So plane x+y=δ(2,1,-1) satisfies if δ=3

Q. 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 $αx + β y + γ z = δ$ . Then which of the following statements is/are TRUE?

$α + β = 2$

$δ - γ = 3$

$δ + β = 4$

$α + β + γ = δ$

Direction ratio of line joining $(3, 2, - 1) & (1, 0, - 1)$ is $(2, 2, 0)$ proportional to $(α, β, γ)$
$\Rightarrow γ = 0, α + γ = 1, α = 1, β = 1$
So plane $x + y = δ (2, 1, - 1)$ satisfies if $δ = 3$

Q. Let α,β,γ,δ be real numbers such that α2+β2+γ2=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 αx+βy+γz=δ. Then which of the following statements is/are TRUE?

α+β=2

δ−γ=3

δ+β=4

α+β+γ=δ