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NDA 2020 Maths · Mathematics · Three Dimensional Geometry · hard

The foot of the perpendicular from the origin to the plane 2x+y+2z=18 is

Single correct option

The foot of the perpendicular from the origin to the plane 2x+y+2z=182x+y+2z=18 is
  1. A. (4,2,4)(4,2,4)
  2. B. (2,1,2)(2,1,2)
  3. C. (6,0,3)(6,0,3)
  4. D. (9,0,0)(9,0,0)
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Answer: (A) (4,2,4)(4,2,4)

The foot lies along the normal (2,1,2)(2,1,2), so it is (2k,k,2k)(2k,k,2k). Substituting: 4k+k+4k=184k+k+4k=18, so k=2k=2 and the foot is (4,2,4)(4,2,4). The point (2,1,2)(2,1,2) is the normal vector itself and is not on the plane, while (6,0,3)(6,0,3) and (9,0,0)(9,0,0) do lie on the plane but are not along the normal from the origin. The distance is 18/3=6=16+4+1618/3=6=\sqrt{16+4+16}, a useful check.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake