Question

Let S be the set of all real values of λ such that a plane passing through the points (–λ²,1,1), (, –λ², 1) and (1,1, –λ²) also passes through the point (–1, –1, 1). Then S is equal to:
(A) {√3} (B) {√3, –√3}
(C) {1, –1} (D) {3, –3}

Answer 1

Solution:

Let A(x₁ , y₁ ,z₁) , B(x₂ , y₂ ,z₂) , C(x₃ , y₃ ,z₃)

then the plane is given by

$\begin{vmatrix}x - x_{1} & y - y_{1} & z - z_{1} \\ x - x_{2} & y - y_{2} & z - z_{2} \\ x - x_{3} & y - y_{3} & z - z_{3} \\ \end{vmatrix} = 0$

$\begin{vmatrix}-\lambda^{2} - 1 & 1 + \lambda^{2} & 0 \\ - 1 -\lambda^{2} & 0 & 1 + \lambda^{2} \\ -\lambda^{2} + 1 & 2 & 0 \\ \end{vmatrix} = 0$

-λ²-1 [(0)(0) - (1+λ²)(2)] - (1+λ²)[(-1-λ²)(0) - (1+λ²)(-λ²+1)] + 0[(-1-λ²)(2) - (0)(λ²+1) = 0

0 + 0 + (λ² + 1)² (1-λ²) - [0 + 2(1+λ²)(-λ²-1) + 0] = 0

(1+λ²)²(1-λ²) + 2(1+λ²)(1+λ²) = 0

(1+λ²)²[1-λ²+2] = 0

λ² = 3

λ = ±√3

(B){√3, –√3} will be the correct answer.