JEE Main 2021 · 24 February · evening · Single correct · +4 / −1

Let a, b∈R. If the mirror image of the point P(a, 6, 9) with respect to the line
(x - 3)/7 = (y - 2)/5 = (z - 1)/(- 9) is (20, b, -a-9), then | a + b |, is equal to :

  1. (a) 88
  2. (b) 90
  3. (c) 86
  4. (d) 84
Answer and solution

Answer (a) 88

Solution

Maths sir · 4 steps

The midpoint of P and its image lies on the line: write it with one parameter and match coordinates.

  1. Midpoint on line. The line is the perpendicular bisector of PQ, so the midpoint M of P(a, 6, 9) and Q(20, b, -a-9) lies on it. Any point of the line is (3+7λ, 2+5λ, 1-9λ).

    M = ((a+20)/2, (b+6)/2, -a/2) = (3+7λ, 2+5λ, 1-9λ)

  2. Match coordinates. The x-coordinates give a = 14λ - 14. The z-coordinates give a = 18λ - 2. Equating them gives 4λ = -12, so λ = -3 and a = -56.
  3. Find b and check. The y-coordinates give b = 10λ - 2 = -32. Check that PQ = (76, -38, 38) is perpendicular to (7, 5, -9): 532 - 190 - 342 = 0, so it is.
  4. So. a + b = -56 - 32 = -88, so |a + b| = 88, which is option (a).

The idea to keep. For the image of a point in a line, put the midpoint on the line using λ. Matching coordinates often fixes everything, and the perpendicular condition is your check.

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

Study Three Dimensional Geometry

Asked about 300 times in JEE Main since 2020. This question is from 2021.