JEE Main 2020 · 8 January · evening · Single correct · +4 / −1

The mirror image of the point (1, 2, 3) in a plane is
( - 7/3, - 4/3, - 1/3 ). Which of the following points lies on this plane ?

  1. (a) (1, –1, 1)
  2. (b) (–1, –1, –1)
  3. (c) (–1, –1, 1)
  4. (d) (1, 1, 1)
Answer and solution

Answer (a) (1, –1, 1)

Solution

Maths sir · 3 steps

The plane is the perpendicular bisector of the point and its image, so find the midpoint and the normal.

  1. Normal to the plane. The line joining P(1, 2, 3) to its image Q(-7/3, -4/3, -1/3) is perpendicular to the plane. Its direction is Q - P = (-10/3, -10/3, -10/3), which is parallel to (1, 1, 1).
  2. Point on the plane. The midpoint of PQ lies on the plane. It is ((1 - 7/3)/2, (2 - 4/3)/2, (3 - 1/3)/2) = (-2/3, 1/3, 4/3).

    x + y + z = -2/3 + 1/3 + 4/3 = 1

    OptionPointx + y + zOn the plane?
    (a)(1, -1, 1)1yes
    (b)(-1, -1, -1)-3no
    (c)(-1, -1, 1)-1no
    (d)(1, 1, 1)3no
  3. So. The plane is x + y + z = 1, and only (1, -1, 1) satisfies it. The answer is (a).

The idea to keep. A point and its mirror image fix the plane completely: the joining line gives the normal, and the midpoint gives a point on the plane.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 300 times in JEE Main since 2020. This question is from 2020.