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← JEE Advanced 2020 · Paper 2
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JEE Advanced 2020 · Paper 2, Mathematics question 10

Let α,β,γ,δ\alpha, \beta, \gamma, \delta be real numbers such that α2+β2+γ2≠0\alpha^2 + \beta^2 + \gamma^2 \ne 0 and α+γ=1\alpha + \gamma = 1. Suppose the point (3,2,−1)(3, 2, -1) is the mirror image of the point (1,0,−1)(1, 0, -1) with respect to the plane αx+βy+γz=δ\alpha x + \beta y + \gamma z = \delta. Then which of the following statements is/are TRUE?

Choose one or more options
Answer and worked solution

Official answer: A, B, C

Normal direction. The segment from (1,0,−1)(1,0,-1) to (3,2,−1)(3,2,-1) is along (2,2,0)(2,2,0), so the plane's normal is parallel to it: α=β\alpha=\beta, γ=0\gamma=0.

Coefficients. α+γ=1\alpha+\gamma=1 gives α=β=1\alpha=\beta=1.

Constant. The midpoint (2,1,−1)(2,1,-1) lies on the plane, so δ=2+1+0=3\delta=2+1+0=3.

Checks.

  • (A) α+β=2\alpha+\beta=2. True.
  • (B) δ−γ=3\delta-\gamma=3. True.
  • (C) δ+β=4\delta+\beta=4. True.
  • (D) α+β+γ=2≠3\alpha+\beta+\gamma=2\ne3. False.

(A), (B), (C)

Topics:Equation of a plane
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