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  4. 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?

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

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Solution:

Direction ratio of line joining $(3,2,-1) \&(1,0,-1)$ is $(2,2,0)$ proportional to $(\alpha, \beta, \gamma)$
$\Rightarrow \gamma=0, \alpha+\gamma=1, \alpha=1, \beta=1$
So plane $x+y=\delta(2,1,-1)$ satisfies if $\delta=3$