If $(1,5,35),(7,5,5),(1, \lambda, 7)$ and $(2 \lambda, 1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Three Dimensional Geometry

(1, 5, 35), (7, 5, 5), (1, λ, 7)

and

(2 λ, 1, 2)

are coplanar, then the sum of all possible values of

λ

is :

(A) $- \frac{39}{5}$
(B) $\frac{39}{5}$
(C) $\frac{44}{5}$ ✓ Correct
(D) $- \frac{44}{5}$

Answer: (C)

Solution

Take

A (1, 5, 35)

as base point, so that

A B = (6, 0, - 30), A C = (0, λ - 5, - 28), A D = (2 λ - 1, - 4, - 33)

The four points are coplanar when the scalar triple product of these vectors vanishes:

60 2 λ - 1 0 λ - 5 - 4 - 30 - 28 - 33 = 0

Expanding along the first row,

6 [- 33 (λ - 5) - 112] - 30 [- (λ - 5) (2 λ - 1)] = 60 λ^{2} - 528 λ + 468 = 0

that is

5 λ^{2} - 44 λ + 39 = 0

, whose discriminant

1156

is positive, so both roots are real.
The sum of all possible values of

λ

\frac{44}{5}

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Single correct option

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