Let S be the set of all real values of $\lambda$ such that a plane passing through the points $\left(-\lambda^{2}, 1,1\right),\left(1,-\lambda^{2}, 1\right)$ and $\left(1,1,-\lambda^{2}\right)$ also passes through the point $(-1,-1,1)$. Then $S$ is equal to:

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Three Dimensional Geometry

Let S be the set of all real values of

λ

such that a plane passing through the points

(- λ^{2}, 1, 1), (1, - λ^{2}, 1)

and

(1, 1, - λ^{2})

also passes through the point

(- 1, - 1, 1)

. Then

S

is equal to:

(A) ${3}$
(B) ${3, - 3}$ ✓ Correct
(C) ${1, - 1}$
(D) ${3, - 3}$

Answer: (B)

Solution

Take

(- 1, - 1, 1)

as reference; the other three points are coplanar with it when the determinant of the edge vectors vanishes. Expanding gives

(- λ^{2} - 1) [(1 - λ^{2})^{2} - 4] = 0

. Since

λ^{2} + 1 \neq = 0

(1 - λ^{2})^{2} = 4

, so

1 - λ^{2} = \pm 2

; this yields

λ^{2} = 3

, rejecting

λ^{2} = - 1

, hence

λ = \pm 3

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Single correct option

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