Let S be the set of all real values of λ such that a plane passing through the point (–λ2, 1, 1), and (1, 1, – λ2), (1, –λ2, 1) also passes through the point (–1,

Engineering

Mathematics

Coplanarity of Four Points

theorem in space

Distance from a Plane

Question

Let S be the set of all real values of λ such that a plane passing through the point (–λ², 1, 1), and (1, 1, – λ²), (1, –λ², 1) also passes through the point (–1, –1, 1). Then S is equal to

${\sqrt{3}, - \sqrt{3}}$

(3, 0)

$(- \sqrt{2}, 1)$

${\sqrt{3}}$

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${[\begin{matrix} \vec{a} - \vec{d} & \vec{b} - \vec{d} & \vec{c} - \vec{d} \end{matrix}]}^{} = 0$

$| \begin{matrix} 1 - λ^{2} & 2 & 0 \\ 2 & 1 - λ^{2} & 0 \\ 2 & 2 & - λ^{2} - 1 \end{matrix} | = 0$

⇒ (λ² + 1) ((1 – λ²)² – 4) = 0

⇒ 1 – λ² = ± 2 ⇒ λ² = 1 ± 2 ⇒ λ² = – 1 or 3

⇒ λ = ±. $\sqrt{3}$