3D Geometry 3 Question 16

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

(2019 Main, 12 Jan II)

(a) 3,3

(b) 3,3

(c) 1,1

(d) 3

Show Answer

Answer:

(a)

Solution:

  1. According to the question points (λ2,1,1),(1,λ2,1) and (1,1,λ2) are coplanar with the point (1,1,1), so

|1λ22021λ2022λ21|=0|x2x1y2y1z2z1x3x1y3y1z3z1x4x1y4y1z4z1|=0(1λ2)[(1λ2)24]=0(1+λ2)[(1λ22)(1λ2+2)]=0(1+λ2)2(3λ2)=0λ2=3[1+λ20λR]λ=±3



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