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NDA 2023 Maths · Mathematics · Three Dimensional Geometry · easy

A line makes equal angles with the three coordinate axes. Its direction cosines can be

Single correct option

A line makes equal angles with the three coordinate axes. Its direction cosines can be
  1. A. 13,13,13\dfrac13,\dfrac13,\dfrac13
  2. B. 12,12,12\dfrac12,\dfrac12,\dfrac12
  3. C. 1,1,11,1,1
  4. D. 13,13,13\dfrac1{\sqrt3},\dfrac1{\sqrt3},\dfrac1{\sqrt3}
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Answer: (D) 13,13,13\dfrac1{\sqrt3},\dfrac1{\sqrt3},\dfrac1{\sqrt3}

Equal angles mean l=m=nl=m=n, and l2+m2+n2=1l^2+m^2+n^2=1 gives 3l2=13l^2=1, so l=±13l=\pm\dfrac1{\sqrt3}. The sum of squares in the other options is 13\tfrac13, 34\tfrac34 and 33, not 11, so none of them can be direction cosines.
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