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}
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Solution:
Let A(x₁ , y₁ ,z₁) , B(x₂ , y₂ ,z₂) , C(x₃ , y₃ ,z₃)
then the plane is given by
-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.
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